Much of my answer is similar to the answer to the question, "Is there any Biblical basis for believing in the Book of Mormon?" and many of the passages will overlap.
The Joseph Smith Translation, a revision of the KJV, contains a clear prophecy about Joseph Smith, and even refers to him by name. This prophecy is found however in an addition to Genesis 50, for which there is no textual support in either the Hebrew text or the Septuagint (nor is it found in any other English translation of Genesis.) While Mormon believers may use these verses, non-LDS Christians will deny their authenticity.
And that seer will I bless, and they that seek to destroy him shall be confounded; for this promise I give unto you; for I will remember you from generation to generation; and his name shall be called Joseph, and it shall be after the name of his father; and he shall be like unto you; for the thing which the Lord shall bring forth by his hand shall bring my people unto salvation. (Genesis 50:33)
There's also significant detail in Isaiah which the LDS believe were fulfilled in the events related to Joseph Smith. It begins about the Great Apostasy, which is the long period of time between Christ and the Restoration during which the fulness of the gospel was not on the earth. The rest of the passage's fulfillment is found in JS-H 1:11-26,63-65.
10 For the Lord hath poured out upon you the spirit of deep sleep, and hath closed your eyes: the prophets and your rulers, the seers hath he covered.
11 And the vision of all is become unto you as the words of a book
that is sealed, which men deliver to one that is learned, saying, Read
this, I pray thee: and he saith, I cannot; for it is sealed:
12 And the book is delivered to him that is not learned, saying, Read
this, I pray thee: and he saith, I am not learned.
13 ¶Wherefore the Lord said, Forasmuch as this people draw near me
with their mouth, and with their lips do honour me, but have removed
their heart far from me, and their fear toward me is taught by the
precept of men:
14 Therefore, behold, I will proceed to do a marvellous work among
this people, even a marvellous work and a wonder: for the wisdom of
their wise men shall perish, and the understanding of their prudent
men shall be hid.
15 Woe unto them that seek deep to hide their counsel from the Lord,
and their works are in the dark, and they say, Who seeth us? and who
16 Surely your turning of things upside down shall be esteemed as the
potter’s clay: for shall the work say of him that made it, He made me
not? or shall the thing framed say of him that framed it, He had no
17 Is it not yet a very little while, and Lebanon shall be turned
into a fruitful field, and the fruitful field shall be esteemed as a
18 ¶And in that day shall the deaf hear the words of the book, and
the eyes of the blind shall see out of obscurity, and out of darkness.
Then there's Ephesians, which mentions this period of time:
10 That in the dispensation of the fulness of times he might gather
together in one all things in Christ, both which are in heaven, and
which are on earth; even in him:
The LDS believe this is the dispensation of the fulness of times, or the time when the gospel is "dispensed" in its fulness, in that all things are restored to the Earth in preparation for the second coming of the Messiah. This verse also references the gathering of the house of Israel, which in our day, is more plainly called missionary work.
Acts 3:21 also mentions the "restitution" or restoration of all things:
21 Whom the heaven must receive until the times of restitution of all things, which God hath spoken by the mouth of all his holy prophets since the world began.
Evidently, "all His holy prophets" taught about the restitution of all things. And apparently, we have lost some of their writings, since in the Bible we have today, not all the prophets specifically mention this thing.