The former England boss, whose side were beaten by QPR in the Championship play-off final last weekend, has been linked with the vacant post at Southampton.
McClaren left a coaching role at QPR to sign a two-and-a-half-year deal at Pride Park last September, revitalising a faltering team and placing them on the cusp of a return to the Premier League.
And Rush told Sky Sports: “I’m delighted with what Steve has done and I think he’s happy here.
“I would be very resistant if anyone came knocking and I wouldn’t advise them to.
“He’s only been here for a few months and I was sure he’d be successful. I’m looking forward to working with him long into the future.
“He’s at the start of his project here and he’s looking forward to get cracking with a big summer. It will be his first pre-season at Derby.”
West Bromwich Albion are the only other Premier League club currently still looking for a manager after Mauricio Pochettino left Southampton to join Tottenham on Tuesday.
McClaren is 14/1 with Sky Bet to take over at St Mary’s, despite Rush seeming to take his club’s head coach out of the running, with current Feyenoord boss Ronald Koeman the 7/2 favourite.
Ex-Basel boss Murat Yakin and former Swansea manager Michael Laudrup are also high on a list that includes out-of-work trio Neil Lennon, David Moyes and Malky Mackay.