The Premiership giants have paid £1m up front for the 22-year-old, who never made a first team start for the Championship side.
The remainder, it is believed will follow if the talented youngster breaks into the United first team and goes on to win international honours.
John Rudge, Stoke's Director of Football said: "Ben has the potential to do well at United and we wish him the very best of luck."
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Foster joined Stoke's academy in August 2002 and spent much of his time in the youth and reserve sides or being farmed out on loan, his last loan deal - at Wrexham - where he was immense between the sticks for the north Wales club effectively lead to his permanent move away from the Potters, with Sir Alex Ferguson tipped off by his son Darren.
Rudge added: "It's an incredible deal which adds further stability to the club and we couldn't stand in his way once we heard Manchester United wanted to sign him."