The Times detailed the Toffees' proposal, with Wilshere facing an uncertain future (h/t MailOnline's James Dutton). His contract with the Gunners expires in the summer, and Dutton pointed out how "Wilshere wants to stay at Arsenal but is unwilling to accept the pay cut currently on the table."
As Dutton noted, Wilshere has received one contract offer from the Gunners despite re-establishing himself in the first team this term and even earning the captain's armband on occasion.
Wilshere has made 17 appearances in the Premier League, as well as becoming a feature of the north London club's UEFA Europa League campaign.
Wilshere has contributed a goal and an assist to help Arsenal reach the last eight in Europe. Manager Arsene Wenger has often deployed Wilshere as a No. 10 during the European run, taking advantage of the 26-year-old's flawless technique and creative verve.
Those qualities recently saw Wilshere earn a recall to the England squad, per BBC Sport.
They are also the same traits Everton are keen to acquire. The Merseyside club has been heavily linked with Wilshere recently, and Sami Mokbel of the Daily Mail reported the Toffees want to pay Wilshere £80,000 per week.
Wilshere said earlier in March there hasn't been much movement on talks aimed at extending his stay with the Gunners, per James Olley of the London Evening Standard.
Olley detailed how Arsenal are offering less than the player's existing £90,000-per-week salary, although the deal would be incentivised. Those sweeteners make sense given Wilshere's career has been interrupted by several injuries.
When fit, though, the England man remains a key player for Wenger. His style and vision are core attributes in the possession-based, attacking game the Frenchman loves his team to play.
The Arsenal boss will also be mindful about the other potential holes set to appear in his midfield ranks. Aaron Ramsey's contract is up in 2019, and London rivals Chelsea are showing interest in the dynamic Welshman, according to MailOnline's Alex Martin.
Arsenal's dilemma will be whether to match the type of lucrative offer Everton are considering making for a player with Wilshere's lengthy injury history.