Toffees boss Marco Silva is on the look out for a new midfielder, and has been credited with an interest in Wilshere, who will leave the Gunners on a free transfer.
The 26-year-old had previously accepted a reduced-terms deal to extend his contract which expires at the end of the month but discussions with Arsenal boss Unai Emery convinced him he had 'little choice' but to leave for "footballing reasons".
Wilshere and his agent had been negotiating his future with the club for several months but the meeting with Emery, appointed as Arsene Wenger's successor in late May, proved the turning point.
"I can confirm I will be leaving Arsenal Football Club when my contract expires at the end of June," Wilshere wrote on Instagram. "Following a number of extensive conversations with those at the club, and in particular a recent meeting with the new manager Unai Emery, I felt that I was ultimately left with little choice but to make the decision that I have due to purely footballing reasons."
Everton will have to move quickly if they wish to do a deal as West Ham, Wolves, Southampton, AC Milan and Juventus are among the interested clubs.