The 26-year-old will be a free agent at the end of the season, and contract negotiations with his current club have so far led to nothing. In an interview on TalkSport's Alan Brazil Sports Breakfast, Allardyce didn't hide the fact he's a fan of the player:
"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."
According to John Cross in the Mirror, the Gunners are unwilling to improve the current contract offer they have on the table, leading to a standoff between the two parties. Wilshere has confirmed talks are going nowhere, per the Press Association's Mark Mann-Bryans:
Cross' report linked Wilshere with a host of interested clubs, including the Toffees, Liverpool, Manchester City, Juventus and AC Milan. He believes letting Wilshere walk would be a mistake on Arsenal's part:
Wilshere would have to take a pay cut to stay at Arsenal and sign a deal laced with incentives, per Cross, likely to counter his lengthy history with injuries. His many health concerns stunted his development over the years and even led to a season on loan at Bournemouth.
Per Transfermarkt, he's mostly managed to stay healthy this season, and a spell of good form saw him rise in the Gunners' midfield pecking order once again. He also returned to the national team setup, but just as things were going well, it was announced he would miss England's friendly against Italy with a knee problem after missing the 1-0 victory over the Netherlands.
The latest setback could hurt his case in the ongoing contract talks. Arsenal appear willing to let him walk if he doesn't give in to their demands, and Wilshere has given no indication he will do so.
Everton could be in for another overhaul during the summer following a disappointing 2017-18 campaign, and a player of Wilshere's calibre would certainly be welcome. Whether he feels the same way remains to be seen, particularly if top clubs from Italy and England are interested.