Blackburn manager Gary Bowyer says Hull have made a bid for prolific striker Jordan Rhodes.
Angel Di Maria has told Real Madrid he wants to leave the club and Sami Khedira could also be on the move.
Ireland, 27, follows Austrian international forward Marko Arnautovic to the Britannia Stadium after he completed his move from Werder Bremen.
Stoke boss Mark Hughes managed Ireland at Manchester City and the club's chief executive Tony Scholes hopes the midfielder can recapture the form he showed then.
Scholes said: "Stephen is a player that Mark knows really well and when the opportunity arose he was keen to add him to his midfield and attacking options.
"Arguably the most successful period of Stephen's career to date was when he played for Mark at Manchester City and we hope that he can rediscover that form."