Wilshere is looking to win a new deal with the North London outfit but despite making 31 appearances in all competitions, the two parties are yet to come to an agreement and the England international has just over three months remaining on his contract.
Everton have recently been linked with a summer approach for the 26-year-old, and Allardyce has acknowledged that he has always been an admirer of the long-serving Gunner.
The 63-year-old 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."
Wilshere has just been called up to the England squad, with the player in line to make his first international outing since Euro 2016.