The 30-year-old has spent the last five seasons at the Den but he announced last week that he would be on the move when his contract expires in July, taking to Twitter to thank the club for giving him a chance in the Football League on the back of a prolific spell at FC Halifax.
Gregory scored 10 Championship goals for Millwall in 2018/19 and, according to the Mail, those displays caught the eye of Sheffield United, who sealed their return to the Premier League recently after 12 years out of the top flight.
A Derby side who will join the Blades in the first division if they beat Aston Villa in the play-off final at Wembley are also interested.
But the Mail claims that he has instead agreed terms to join Stoke on a free transfer, with the deal likely to be completed by the end of the week.
This is despite the fact that the Potters finished 15th in the Championship, 19 points adrift of the play-off places.
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