The People say the Turkish giants submitted offers for a permanent or loan deal.
But their swoop came too late and a deal failed to materialise.
Fenerbahce did however manage to snap up Vincent Janssen from Tottenham on loan.
Wilshere had been targeted by several clubs but ended up remaining in north London.
Serie A outfit Sampdoria were keen along with Championship sides Aston Villa and Birmingham.
In the Premier League, Wilshere was eyed by West Ham, Swansea and Newcastle.
Former player turned pundit Tony Cascarino reckons the England international is at a crossroads in his career and should have moved abroad.
He told The Times: “He seemed to join Bournemouth because his friend Benik Afobe was there, rather than pick a club where he was going to prove himself.
“Eddie Howe saw that he needed to play three in midfield to accommodate Wilshere and, midway through the season, the England man was on the bench.
“Howe wasn't in a hurry to re-sign him as he did with Nathan Ake, who was on loan from Chelsea and then joined permanently for £20m.
“Wilshere should have moved abroad —despite the language barrier, the slight isolation forces you to focus on football.”
Wilshere, 25, was sent out on loan to Bournemouth last season in search of regular match action.
But he underwhelmed at the Cherries.
He made 29 Premier League appearances, failing to score a single goal before breaking his leg in April.
There are just 12 months left to run on his Gunners contract.
