Wilshere has re-established himself in the Arsenal first team this season after years of injury torment and spending last season out on loan at Bournemouth.
However, he can leave on a free transfer unless a new deal is agreed in the next couple of months.
“All I can tell you is that we are no closer to sorting anything out,” Wenger said ahead of Arsenal’s Europa League last 16, second leg at home to AC Milan on Thursday.
“We make a proposal to him, I want him to stay and my desire is to see him linked to the club for the future. I hope he will do it.”
However, reports suggest the offer on the table for Wilshere is for a lower basic salary than his current deal.
Arsenal’s forward planning for players running down their contracts has been much-criticised after Alexis Sanchez joined Manchester United in January.
The Chilean’s market value tumbled as he had just six months left to run on his deal before Arsenal and United agreed to a swap deal with Henrikh Mkhitaryan moving in the other direction.
Arsenal’s bargaining position was also weakened before handing Mesut Ozil a lucrative three-year contract extension in January worth a reported £350,000 a week.
Wilshere may be hoping to prove himself at the World Cup in a bid to impress potential suitors, and looks set for a recall when Gareth Southgate names his England squad for friendlies later this month against the Netherlands and Italy on Thursday.
“The vibes I get are positive. I believe he has a good chance to be in there, but we have to leave that to Gareth Southgate,” added Wenger.
“I have enough to do with my job.”
The intense pressure on Wenger to end his 22-year stay after a dreadful run of results was eased slightly by an impressive 2-0 Arsenal win away in Milan in the first leg last week.
Winning the Europa League is Arsenal’s last realistic route into the Champions League next season as they trail fourth-placed Liverpool by 12 points in the Premier League.
“It is a massive opportunity for us,” said midfielder Aaron Ramsey on Arsenal’s prospects of reaching the last eight.
“This game is very important for us and our season.”
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