Grant began his professional career with Derby County making his Football League debut in September 2002.
"Moving to Manchester United has been a dream of mine for as long as I can remember", he told ManUtd.com.
Manchester United are reportedly close to announcing their third signing of the summer transfer window.
That has resulted in the 22-year-old, who Mourinho regards as the best Portuguese goalkeeper of his generation, asking to be allowed out on loan in order to gain more experience.
However, the attacker wasn't able to prevent Stoke from being relegated to the Championship last season after they finished 19th in the Premier League and Shaqiri has made it clear his future now lies elsewhere.
England vs Sweden: everything you need to know
Sweden topped their group, beating Mexico and South Korea, before edging past Switzerland 1-0 to set up clash with England . They've scored more goals (12) than any team at the tournament, while Brazil have conceded the least goals (1).
Coincidentally, Grant was in goal for Stoke when Wayne Rooney scored his record-breaking 250th goal for Manchester United in January 2017, denying the Potters three points and a clean sheet.
Finally, Juventus left-back Alex Sandro has agreed to join United in a deal worth £53m, says the Daily Star.
This meant Mourinho was set to only possess two goalkeepers - De Gea and Romero.
However, the big story involving United on Thursday morning is that they are ready to make a move for a different kind of playmaker.
It could yet be one in the latest of a string of transfers where Shaqiri - who is heavily linked with a move to Liverpool - goes to Anfield with the red herring of United's interest behind him to make it look like Klopp won "another" transfer battle.