Crouch, whose goal in the 2-2 draw with Wigan Athletic on Tuesday night (Wednesday EDT) was his sixth of the season, and first since September, has recently been linked with a return to his former club.
"QPR have enquired about Peter," Stoke manager Tony Pulis said after the Wigan match.
"Whether we do it or not, you will have to wait and see.
"Harry has rung me up and asked me about the possibility. I have spoken to Peter about it and then it is up to Peter whether he wants to go. Peter is very happy here and he has made that clear.
"The important thing for me is that if he does want to go, it has to be right for us and also we have to make sure we have someone who will come in and do the same job."
Redknapp, for whom Crouch has played previously at Portsmouth (twice), Tottenham Hotspur and Southampton, meanwhile said after QPR's home draw with Manchester City: "I love him as a boy. Now it's a question of whether Stoke would sell him. Tony Pulis is a friend of mine, but we'll see."
Stoke rejected an initial bid of £3 million ($4.45 million) plus Jamie Mackie, but Redknapp is reported to be preparing an improved £6 million bid.
Pulis said "it would be nice to freshen a few things up ... we'd like to get a few new faces in".
"And there are players here who are not playing that maybe could go out and play, but it has to be the right deals.
"There has been talk about Peter Crouch, about Kenwyne Jones (the Stoke striker linked with Swansea City), about (United States and FC Dallas winger) Brek Shea, loads of stuff.
"I will do my business the way I always done it."
Redknapp is also reported to want Christopher Samba, the former Blackburn Rovers defender now at Russian club Anzhi Makhachkala.
Samba, 28, rejected the opportunity to join Rangers in the transfer window 12 months ago, but he is now reportedly keen to return to English football.
Rangers have allowed defender Anton Ferdinand to join Turkish club Bursaspor on loan for the rest of the season, having conceded 37 goals in the league this season, and Anzhi are report to be considering an offer of £8 million.
Such a deal would represent a club record, after Rangers recently signed Loic Remy from Marseille for £8 million.