Wilshere is running into the final months of his deal at the Emirates, and the Mirror reports that the Toffees are ready to put an offer on the table than would be better than the one that Arsenal are currently offering the £110,000-a-week star.
The Mirror hints that it could be tough for Everton to win the race for the Englishman's signature however, with Wilshere keen to stay with the Gunners and Italian giants Juventus also contemplating a move for the 26-year-old.
The Daily Mirror reported earlier in the month that Everton are considering a move for their former midfielder if they decide to get rid of Sam Allardyce following what has been an underwhelming tenure at Goodison Park.
Everton are, of course, not the only top flight team in England that Arteta represented during his playing days as he also played 149 times for Arsenal before hanging up his boots.
Many of those appearances would have been alongside Wilshere, so perhaps their previous working relationship may make Everton a much more tempting proposition for the midfielder if the Toffees do end up going after Arteta.
Wilshere has been a shining light for Arsenal this season, so getting him through the door at Everton would be a huge coup for the Merseyside club if they can pull it off.