Wilshere made 27 league appearances under Eddie Howe last term before suffering a leg break in the Cherries' 4-0 defeat at Tottenham in April.
Wilshere has 12 months left to run on his current deal with no agreement over an extension in sight.
That tally made the 2016-17 campaign Wilshere's most successful-or at least most heavily involved-season since 2013-14, having struggled with recurring injuries in recent years.
And though Wenger has moved to rule out a move for Wilshere, the Gunners boss has warned he faces a battle to return to first-team contention.
The England global made 29 Premier League appearances for the Cherries, but his progress was cut short by a fractured leg he suffered in April.
As a result, the midfielder did not travel with the Arsenal squad on their pre-season tour of Australia and China as he stayed at the club's training ground to continue his rehabilitation.
West Ham owner David Gold was the latest Premier league chief to rule out a move as the Hammers supremo suggested his priorities lay elsewhere.