The 25 year old Wilshere has only played 2 Premier League appearances so far this season and Gareth Southgate did not name him in his latest squad for friendlies against Germany and Brazil as a result of the lack of game time which he has seen.
Wilshere has won 34 caps for England, including in the 2014 World Cup, but has not featured for Southgate's side since they were eliminated from the 2016 European Championships.
But new West Ham manager David Moyes has been promised funds for January and the club are expected to sign a new midfielder after missing out on William Carvalho last summer.
Arsenal midfielder Jack Wilshere on Thursday asserted his club is aiming to defeat Tottenham Hotspur in their English Premier League (EPL) tie on Saturday and finish above them in the points table. "That is not the way we are playing".
Wilshere has found his opportunities limited at Arsenal this season after spending last season out on loan at Bournemouth. When we have played with a 10, we have had Dele [Alli] and [Adam] Lallana.
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"I want to be playing for my country", he said.
Wilshere has impressed in the Europa League to help Arsenal make an unbeaten start to their first campaign in Europe's second tier competition with three victories and one stalemate.
I think he is a talent and I really respect him as an individual. You'd have to ask Arsene [Wenger] why that was. I work with [Palace sporting director] Dougie Freedman and we talk about potential players, and Jack Wilshere's name has never been mentioned in this room.
"Now you are telling me we have been linked with him, but by who?"
"We need him both internationally and for the Arsenal", the former captain added. He said I needed more football so that's what I've got to do.