According to David Hytner of the Guardian, the Gunners have offered Wilshere fresh terms worth £80,000 per week but are unwilling to improve that amount to fall in line with what the midfielder wants to earn.
Hytner reported Arsenal's offer is of a "take-it-or-leave-it" nature and that Everton, West Ham United and Juventus are leading the race to land the England international if he rejects his boyhood club's proposal.
As Phil Kirkbride of the Liverpool Echo noted, Everton manager Sam Allardyce commented on the speculation on Friday:
"Wilshere has managed to change [Arsene] Wenger's mind about him to the extent that he does now have the offer of a contract, albeit on reduced basic terms of £80,000 per week, which he finds unacceptable," Hytner said. "But the question has nagged away as to how much Arsenal truly want him to stay at the club."
In a season of numerous low points for the Gunners, Wilshere's return to fitness and form has been one of the big positives for supporters.
Earlier in the campaign, Wilshere mainly featured in the UEFA Europa League. But after some impressive performances in the group stage of the competition, fans clamoured to see the midfielder back in the side for Premier League matches too.
Now, when fit, Wilshere is a key man in this Arsenal team, and with that in mind, losing him for free at the end of the season would leave a void. As Charles Watts of Football.London noted, a club legend rates him highly too:
Although it's unlikely Wilshere will live up to the immense potential he showcased when he first burst on to the scene at Arsenal in 2010, he's still got a lot to offer.
It's difficult to think of an English midfielder with better technical ability, and he has a rare ability to break a game open with a change of pace, a drop of the shoulder or an incisive pass.
Still, the injury issues are a big concern. The Monkey Sponge Twitter account believes teams should be wary of making Wilshere lucrative offers:
You sense there will be a lot of teams prepared to take that gamble, though, especially as they wouldn't have to pay any transfer fee to get the playmaker on board for the 2018-19 season.
Given his affinity with Arsenal and the improvement he's showcased this term, Gunners supporters will hope a resolution can be reached with Wilshere before the end of the campaign. However, it's easy to see the prospect of a better contract and a new challenge luring the player away from the Emirates Stadium.