There is, among many groups, a basic idea that people in the Old Testament, namely Israelites, went to heaven.
To further explain the scope of this question, I want to ignore the specific individuals that were brought up to heaven. They are special cases. I am wondering whether or not there is any scriptural support for the idea that an Israelite went to heaven after death or that salvation (from eternal death) was offered for them.
It's not entirely 'fair' to rule out the few individual O.T. persons who clearly did not just die and rot in their graves. But, for the sake of answering your specific question, I would quote these O.T. scriptures that are self-explanatory in showing the basis for the general idea that conscious awareness after physical death was a consideration, even in O.T. times.
“You guide me with your counsel, and afterwards you will take me into glory.” (Psalm 73:24)
“But God will redeem my life from the grave; he will surely take me to himself.” (Psalm 49:15)
"The days of our years are threescore and ten...or if fourscore yet their strength is labour and sorrow, for it is soon cut off and away we fly." (Psalm 90:10)
“Do not put your trust in princes, in mortal men who cannot save. When their spirit departs they return to the ground” (Psalm 146:4)
“Remember God… before the dust returns to the ground it came from, and the spirit returns to God who gave it… For God will bring every deed into judgment, including every hidden thing, whether it is good or evil.” (Ecclesiastes 12:1-14)
“Multitudes who sleep in the dust of the earth will awake, some to everlasting life, others to shame and everlasting contempt… You will rest, and then at the end of the days you will rise to receive your allotted inheritance.” (Daniel 12:2-13)
“I know that my Redeemer lives, and that in the end he will stand upon the earth. And after my skin has been destroyed, yet in my flesh I will see God; I myself will see him with my own eyes.” (Job 19:25-26)
Now, the following point is important if we are to understand the mind-set of ancient Jewish people. The Hebrew scriptures had two words, qe’ver (grave) and sheol (to which the grave was the entrance). Sheol is often translated as hell. The Rabbinic schools of Shammai and Hillel taught that sheol had two compartments, hell and ‘the Bosom of Abraham’, with a great chasm separating them. This holding place was for the departed spirits of the dead, but on the Day of the Resurrection, they would come forth to be resurrected and judged. Jesus, a Jewish rabbi, agreed with those aspects of Judaic belief as can be seen from His description of hell, and the bosom of Abraham, in the warnings he gave to lovers of money (for the rich man who died ended up in agonizing torment in hell).
This belief in awareness after physical death of being in some aspect or other of God's realm, would not need to be in God's presence, given that it was only after Jesus' resurrection and ascension to heaven that heaven was opened up for others. Peter, after all, said that David's sepulchre was still there, with David's corpse in it, and "David is not ascended into the heavens" yet Peter went on to show that David prophesied about the death and resurrection of Messiah, whose body would not be allowed to corrupt in the grave. David said:
"I foresaw the Lord always before my face, for he is on my right hand, that I should not be moved. Therefore did my heart rejoice, and my tongue was glad; moreover also my flesh shall rest in hope, because thou wilt not leave my soul in hell [the grave] neither wilt thou suffer thine Holy One to see corruption." Psalm 16:8-11 as quoted in Acts 2:24-36 [Emphasis mine]
For confirmation that the O.T. saints who had faith (like Noah, Abraham and Moses) also believed there was life after death, you need to read the whole of Hebrews chapter 11 - too much for me to put down here - but look for phrases like 'a better country, that is, an heavenly' and 'that they might obtain a better resurrection'. However, your other question about "the idea that an Israelite went to heaven after death or that salvation (from eternal death) was offered for them" is fundamentally flawed as you try to make an unwarranted connection between that and (all?) Israelites in general 'going to heaven'. Hebrews chapter 11 will partially address that as it shows the need for faith, not merely being born an Israelite.
See
What about Muslims & Jews? Will they not get salvation?
Daniel 12:2 And many of them that sleep in the dust of the earth shall awake, some to everlasting life, and some to shame and everlasting contempt.
Daniel 12:3 And they that be wise shall shine as the brightness of the firmament; and they that turn many to righteousness as the stars for ever and ever.
This is exactly what Jesus taught the Disciples:
Matthew 25:46 And these shall go away into everlasting punishment: but the righteous into life eternal.
And you know the Israelites [among all nations] were chosen by God for a special reason that Abraham is their father, and what confuses is the question seeks a Hebrew biblical source for the forgiveness of Hebrews from a book that was transmitted along Hebrew generations!
Are you saying God cursed the people who gave you the Scripture?
Exodus 24:7 And he took the book of the covenant, and read in the audience of the people: and they said, All that the LORD hath said will we do, and be obedient.
These people promised to follow the Torah, why will God not show mercy to the obedient?
No, He can, and the proof is Jesus didn't come to abolish the law, nor God left the people after Adam without guidance:
Matthew 5:17 Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil.
To fulfill what Moses [and the Prophets] didn't speak about, which explains why the Jews have a thin knowledge of the Afterlife [unlike Christians], because the law was not complete without the Gospel.
Luke 22:30 That ye may eat and drink at my table in my kingdom, and sit on thrones judging the twelve tribes of Israel.
The people whom you think didn't find salvation had laws prescribed for them to repent from sins by God Himself, the Israelites didn't need a subordinate when they had a messenger helping them walk on God's path:
Genesis 17:1 The LORD appeared to Abram, and said unto him, I am the Almighty God; walk before me, and be thou perfect.
And please, don't let the sarcastic question in the beginning anger you, but portraying the entire Israelites as damned without chance of God's mercy means you misunderstood God's justice, and even Christ's message:
Matthew 8
10 When Jesus heard it, he marvelled, and said to them that followed, Verily I say unto you, I have not found so great faith, no, not in Israel.
11 And I say unto you, That many shall come from the east and west, and shall sit down with Abraham, and Isaac, and Jacob, in the kingdom of heaven.
Did the old testament people have hope of salvation? - This is the question I will answer.
Matt 8:11 And I say unto you, That many shall come from the east and west, and shall sit down with Abraham, and Isaac, and Jacob, in the kingdom of heaven.
Jesus, by telling this clarified that Abraham, Isaac and Jacob will be in the Kingdom of heaven.
The author of Hebrews makes it abundantly clear, after giving a list of old testament characters who overcame by faith, he said:
Heb 11:35 Women received their dead raised to life again: and others were tortured, not accepting deliverance; that they might obtain a better resurrection:
And further
Heb 11:39,40 And these all, having obtained a good report through faith, received not the promise: God having provided some better thing for us, that they without us should not be made perfect.
They without us should not be made perfect - clearly means that when we obtain the reward of life, they will obtain it too.
Again,
Heb 11:13-16 These all died in faith, not having received the promises, but having seen them afar off, and were persuaded of them, and embraced them, and confessed that they were strangers and pilgrims on the earth. For they that say such things declare plainly that they seek a country. And truly, if they had been mindful of that country from whence they came out, they might have had opportunity to have returned. But now they desire a better country, that is, an heavenly: wherefore God is not ashamed to be called their God: for he hath prepared for them a city.
God told Daniel:
Dan 12:13 But go thou thy way till the end be: for thou shalt rest, and stand in thy lot at the end of the days.
Daniel will receive his reward at the end of days, in the resurrection of the just.
Job said:
Job 19:25-27 For I know that my redeemer liveth, and that he shall stand at the latter day upon the earth: And though after my skin worms destroy this body, yet in my flesh shall I see God: Whom I shall see for myself, and mine eyes shall behold, and not another; though my reins be consumed within me.
David said:
Ps 17:15 As for me, I will behold thy face in righteousness: I shall be satisfied, when I awake, with thy likeness.
When I awake with Thy likeness - with God's likeness. This happens at resurrection.
1John 3:2 Beloved, now are we the sons of God, and it doth not yet appear what we shall be: but we know that, when he shall appear, we shall be like him; for we shall see him as he is.
Forgiveness of sins were promised to the people of the Old Testament as surely as it is promised to the New Testament people
Isa 1:18 Come now, and let us reason together, saith the LORD: though your sins be as scarlet, they shall be as white as snow; though they be red like crimson, they shall be as wool.
Ps 51:1,2 To the chief Musician, A Psalm of David, when Nathan the prophet came unto him, after he had gone in to Bathsheba. Have mercy upon me, O God, according to thy lovingkindness: according unto the multitude of thy tender mercies blot out my transgressions. Wash me thoroughly from mine iniquity, and cleanse me from my sin.
Ps 51:7-12 Purge me with hyssop, and I shall be clean: wash me, and I shall be whiter than snow. Make me to hear joy and gladness; that the bones which thou hast broken may rejoice. Hide thy face from my sins, and blot out all mine iniquities. Create in me a clean heart, O God; and renew a right spirit within me. Cast me not away from thy presence; and take not thy holy spirit from me. Restore unto me the joy of thy salvation; and uphold me with thy free spirit.
Ps 32:1-6 A Psalm of David, Maschil. Blessed is he whose transgression is forgiven, whose sin is covered. Blessed is the man unto whom the LORD imputeth not iniquity, and in whose spirit there is no guile. When I kept silence, my bones waxed old through my roaring all the day long. For day and night thy hand was heavy upon me: my moisture is turned into the drought of summer. Selah. I acknowledged my sin unto thee, and mine iniquity have I not hid. I said, I will confess my transgressions unto the LORD; and thou forgavest the iniquity of my sin. Selah. For this shall every one that is godly pray unto thee in a time when thou mayest be found: surely in the floods of great waters they shall not come nigh unto him.
David said everyone who is Godly shall pray for this forgiveness of sins in acceptable time. Anyone whose sins are forgiven is just and will partake of the resurrection of the just, as given in Daniel 12:3.
P.S. I do not believe anybody goes to heaven at death. It happens at the resurrection of just except in cases where the Bible specifically says that such a person was taken by God - Examples: Enoch, Elijah.
