Wilshere is out of contract at Arsenal at the end of the season and has been linked with a free transfer to Everton.
Reports claim the Toffees are willing to offer the England international more than the Gunners.
Wilshere has been a regular in the Arsenal side in the Premier League since December and his form has led to him being recalled to the international squad by Gareth Southgate.
Allardyce has also been impressed by the 26-year-old this season but refused to comment on whether he would want to sign him.
“I can’t comment on that, you’re not supposed to be talking about that!,” Sam Allardyce told talkSPORT.
“I’ve always admired him, oh yes.
“I’ve seen a big difference in Jack Wilshere recently, in his frame, in his body.
“He looks like he has really got the bit between his teeth.
“He looks a lot fitter than he was and he is leaner in his face.
“He looks like he’s had a real go off the field, with his diet and extra training, to overcome his injury problems and prove what a great player he is.
“Hopefully if he is over his problems, whether it’s Arsenal or another club, we all hope he has a long career ahead of him.”
