Wilshere listened to their loan offer, but chose to stay at Arsenal and attempt to win his place back in the first team.
He could be in contention to feature in the Europa League tonight against Cologne.
Trabzonspor's approach for Wilshere came after they failed in an attempt to sign Tottenham Hotspur flop Moussa Sissoko.
They made a strong approach to try and sign the France international on loan, but Spurs were insistent they would only let him leave permanently for a big fee.
Wilshere spent last season out on loan at Bournemouth before suffering a late injury which wrecked his pre-season.
His recent return to action in the under-23s ended unhappily as he was sent off for a brawl with a young opponent.
A World Cup squad place seems a long way off for the midfielder right now with Wilshere fighting to earn a place back in Arsenal's team.
The very fact Trabzonspor turned to Moussa Sissoko before they looked at him shows how far his star has recently fallen.