The Sun says Crouch is wanted by Australia's A-League to help raise their profile.
But any deal must be worth his while as the ex-England hitman, 37, hinted he may stay with the Potters in the Championship.
Crouch told Planet Football: "I feel like there is unfinished business for me at Stoke.
"I still have a year left on my contract and while I don't know what is going to happen in the next few weeks, I'd have no problem playing in the Championship next season and trying to get the club back into the Premier League at the first attempt.
"Stoke have a wonderful set of fans, a great set-up in so many areas of the club, and it is a club that should be in the Premier League."