Every true Christian has been called and chosen. (But the example of Judas Iscariot is a warning to us all to make our calling and election sure.)
There is a general call and an effectual call.
Many hear the general call. Whenever a person hears the Gospel with its call to repent and believe on the Son of God, they receive the general call. But many do not respond with repentance and faith for "many are called but few are chosen", Matthew 22:14.
Then there is the effectual call as described by Paul in Romans:
And we know all things work together for good, to them that love God, to them that are the called according to his purpose. For whom he did foreknow, them he also did predestinate to be conformed to the image of his Son, that he might be the firstborn among many brethren.
Moreover whom he did predestinate, them he also called; and whom he called, them he also justified; and whom he justified, them he also glorified. Romans 8:28-30.
The above passage is a golden chain of salvation which cannot be broken. It describes the effectual calling of the few who have been chosen: these are the elect who at some point in life will truly believe, repent, and thus become a true Christian. All true Christians have been effectually called to salvation.
Judas Iscariot was called to be an apostle but was not effectually called to be saved.
Postscript
As Jesus said:
"No one can come to me unless the Father who sent me draws him, and I will raise him up at the last day", John 6:44.
Post postscript
The OP writes:
These two verses show that even for Christians they cannot come to God unless God already called them and, no matter what they do, if God hasn't called them they won't be saved. So is there any circumstance where a believer can have the status of being called but not chosen?
It is really hard to know what is intended here. There are two groups:
There is someone who says he is a Christian but is not, he has not put all his trust in Christ alone; Let me call him a "False Brother". I do not mean he is intentionally false: he might not know he is not a true believer; and
There are those who are truly Christian. Let me call them True Believers.
If in your closing paragraph you mean a True Believer:
A Christian (True Believer) is, by definition, someone who has already come to God.
A Christian is, by definition, already saved.
And a True Believer has come to God and been saved because God has already effectually called them.
If the OP means by "Christian" a False Brother then for as long as others think he is a true Christian he will have the status amongst other believers of having been effectually called but only God knows if he has been chosen.. he may yet be chosen but not yet effectually called.
Is it that "They cannot come to God.. no matter what they do"?
It is not a good way of speaking. It is not that they cannot come to God, it is really that they will not come to God. They have no desire to come to God.
And it is not "no matter what they do". Rather it is that, if they do anything at all, it is not with the sincere desire to find God. They have no desire for God.
If anyone, anywhere pursues the means of faith alone in Christ alone, if anyone pursues the means that God has given in Scripture and calls upon the name of God for help and grace and pardon they will be saved. If anyone seeks, and keeps on seeking, they will find. Our Lord Jesus has promised it.
Though a man may know that he is saved and (if he believes the doctrine of effectual calling and election and predestination) though he may know because he knows he is saved thus he may know he was chosen by God in Christ before the foundation of the world; no man should believe he is not elect such that he makes no effort to find Christ and salvation in Christ. No one knows that they are not elect. So all should pursue the means of salvation. All should believe the words of Jesus Christ "If anyone believes on me... out of his belly shall flow rivers of living water." John 7:38. He should read the scriptures, especially the gospels of Matthew, Mark, Luke, and John.
If anyone knows he is not yet saved, and finds an awakening desire to be saved, he should pursue the means of salvation, faith alone in a crucified Saviour who died that our sins might be forgiven by believing on the only Saviour:
Knock and the door shall be opened to you (Matthew 7:7). It is the promise of the only begotten Son of God.
Look unto me and be you saved all the ends of the earth, for I am God and there is no other. Isaiah 45:22
And just as Moses lifted up the serpent in the wilderness even so must the Son of Man be lifted up that whoever believes in him should not perish, but have eternal life. (Numbers 21:4-9, John 3:14-15)
Come unto me all you that are heavy laden and I will give you rest. Matthew 11:28
To the one who does not work, but believes on him who justifies the ungodly, his faith is credited to him as righteousness. Romans 4:5.
No one should see the doctrine of election as a wall that keeps them out, but once a sinner is saved let him see the doctrine of election be a wall that helps to keep them in. When Christ was dying he was not dying for all the sinners in general, he was dying for his own sheep who he knew by name.
He lovèd me and gave Himself for me,
Amazing love, amazing sacrifice!
I'll take my harp down from the willow tree,
And bid its note in praise of Jesus rise.
He lovèd me and gave Himself for me,
And surely I myself to Him will give;
None Jesus will I ever love like Thee,
And to Thy glory only will I live.