McClean, 29, has signed a four-year contract with the Potters in a deal which could be worth up to £6m with performances based add-ons.
“James is a player I’ve admired for a long time. He has real quality with his left foot and is an old-fashioned winger,” Stoke manager Gary Rowett told the club’s website.
“I’ve spoken to lads who have played with him before, the likes of Darren Fletcher, and they have all said that he’s a great guy to have around the dressing room and he’s really, really popular with his team-mates.”
The Republic of Ireland international is Stoke’s fourth signing of the summer following the arrivals of Benik Afobe, Adam Federici and Oghenekaro Etebo.
Stoke are also interested in defender James Chester and had a £5m bid rejected by Aston Villa for the Wales international last week.
They kick-off their Championship campaign away to Leeds United on August 5.