The 26-year-old’s contract with the Emirates side is up in the summer, and while the Gunners remain optimistic about securing the England man on a new long-term deal, he continues to be linked with a move away on a Bosman.
The player fuelled that speculation by revealing that manager Arsene Wenger told him he was free to leave last summer.
Among the leading pack for the player are the Toffees, with manager Allardyce doing little to quell speculation on Thursday.
“I can’t comment on that, you’re not supposed to be talking about that.”
Allardyce, though, has been impressed with the progress that Wilshere has made physically over the course of the last year.
“I've seen a big difference in Jack Wilshere recently, in his frame, in his body,” he commented. “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.”
Despite the praise of Allardyce, Wilshere has found it harder to win over Wenger, as he has managed little over 1000 minutes of Premier League football this season, spread over 17 matches.
He has one goal and two assists to his name in the league, with the same figures again in nine Europa League outings.