Muslims reject this. The Quran mentions many prophets who are in the Bible, but it often tells the story differently and corrects what Muslims see as changes in the earlier books. Muhammad ﷺ could not read, and no Arabic Bible existed in his time.
The fuller answer
In Mecca of the 7th century, there was no complete Arabic translation of the Bible. The Prophet ﷺ was known to be unlettered, and his people, who opposed him, accused him of getting help from a man, and the Quran answers that the one they point to speaks a foreign language, while this is clear Arabic.
The Quran does not claim to be a copy: it states it confirms what is true in the earlier scriptures and is a criterion over them. It has stories that are not found in the Bible, such as the people of 'Ad and Thamud, and it differs from the Bible on the story of Abraham's building of the Kaaba, the nature of Jesus and others. The similarity is explained, in Islam, by one source: the same God.
And We certainly know that they say, "It is only a human being who teaches the Prophet." The tongue of the one they refer to is foreign, and this Qur'an is [in] a clear Arabic language.
And you did not recite before it any scripture, nor did you inscribe one with your right hand. Otherwise the falsifiers would have had [cause for] doubt.
And We have revealed to you, [O Muhammad], the Book in truth, confirming that which preceded it of the Scripture and as a criterion over it. So judge between them by what Allah has revealed and do not follow their inclinations away from what has come to you of the truth. To each of you We prescribed a law and a method. Had Allah willed, He would have made you one nation [united in religion], but [He intended] to test you in what He has given you; so race to [all that is] good. To Allah is your return all together, and He will [then] inform you concerning that over which you used to differ.
This answer follows the mainstream Sunni understanding and is general learning, not a fatwa. Where Muslims differ, the answer says so. Have a question of your own? Write to us.