That seems to be one of the disconnects in Islam as far as I can tell.
Quran Translated to Modern English. Clear, Pure, Easy to Understand.
www.clearquran.com
Sura 10:
99. Had your Lord willed, everyone on earth would have believed. Will you compel people to become believers?
100. No soul can believe except by God’s leave; and He lays disgrace upon those who refuse to understand.
That is clearly contrary to the idea of free will, souls only believe if Allah wills it. The first line seems to indicate Allah will not force everyone to believe but the second clearly indicates that you may only believe if Allah allows it. If Allah gives us all leave to believe then the second verse is meaningless. It makes no sense in that context. There are many versus exactly like this one, the Quran states over and over and over again that it is Allah who decides what you will and will not believe.
Then in other places it seems to state the exact opposite...
In 16 we get the same story:
93. Had God willed, He would have made you one congregation, but He leaves astray whom He wills, and He guides whom He wills. And you will surely be questioned about what you used to do.
He will lead those that he wishes but leave the others astray. Again, this looks a lot like stating he will make some believe while ignoring others. Hardly a show of free will.