Omitted from the squad for tonight's England game against Germany, the playmaker "is desperate to play for Arsenal".
"You can accelerate people's progress and we saw that with a lot of the players we brought into the England team".
The 25-year-old has mainly been used in the Europa League, starting three of the club's four games in the competition, and has also started twice in the Carabao Cup.
"Jack played as a number 10 all of last season on loan at Bournemouth", Southgate said.
Wenger has played Wilshere as a No 10 this season.
"I have got no idea about where the name of Jack Wilshere has come from, so where did you get it from?"
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"I really like Jack Wilshere as a player", he said.
"I am hoping that we are in a position in March where we can pick him".
"I like him and I think he is a very good player". Southgate holds him in high esteem but his slide down the Arsenal pecking order and the difference of opinion over his deployment means he may need a new club, a new position or both in January. With Hodgson's successor, Gareth Southgate, stating that he can't consider Wilshere because of his lack of game time, such a move would make sense.
"I don't think Jack is a ten", Merson told Sky Sports News. "Of course I want to stay".
Wilshere has made one assist in two substitute appearances in the Premier League but there is an increasing clamour for the Englishman to start ahead of want-away Mesut Ozil.
"I left a year ago to get some football, I've done that and I'm back now and I'm fit and ready to play".