The Yellow Canaries want Wilshere on a January transfer.
Hodgson, 70, will keep an eye on the England international's progress until the New Year and then decide whether or not to swoop.
Fenerbahce could face competition from Trabzonspor for Wilshere.More news: What on earth is this terrifying eyeless sea creature?
Arsenal have dropped out of the Champions League for the first time in 20 years, but Arsene Wenger is still adamant that he would rather that the club return to the Top Four to guarantee Champions League football next season, rather than win the Europa League, like Manchester United did last season.
And Wenger confirmed he will be selected and play for the club for the first time in 13 months. "He needs a little burst to get away from people because he can turn the game forward and if you can turn the game forward you need your legs to get you out of pressure and that will come back - it is coming back in training as well".
"If he can combine the two, keep calm and focus on the job, he is on a good way".More news: Flynn's son is a subject in Russian Federation probe
Alexis Sanchez trained today so he's expected to be in the squad and Wenger may give him the chance to build on his match fitness as he's only started one Arsenal game so far this season - although I think it would be a risk ahead of the Chelsea game.
Europa League glory would guarantee a Champions League place next season - a route taken by Jose Mourinho's Manchester United a year ago - but Wenger believes the best way to qualify is via the English Premier League.
Alternatively Fenerbahce would have to pay a fee if they want him to to sign in Janaury. If he is injury-free, he will come back to his best.More news: The iPhone X, iPhone 8 and more
"I didn't talk to him at all after the red card because I was not at the game and I could understand his reaction".
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