The English midfielder’s current Arsenal deal is due to expire at the end of the season and he was contemplating leaving in search of regular football.
The 25-year-old did not start a Premier League game this season until December 13, but since breaking through he has not missed a league match.
Wenger has always admitted that he is keen to keep hold of Wilshere, especially with the expected departure of Mesut Ozil and Alexis Sanchez.
The pair are out of contract at the end of the season and could leave on the cheap when the winter window reopens in five days’ time.
So the Gunners boss is hoping to tie the former Bolton loanee down on the long term – but they are struggling to agree on the financial terms of the deal.
“We will sit down with him,” Wenger said.
“I believe that if we can meet at a point of agreement financially he will want to stay.”
Arsenal are expected to have a busy transfer window with a number of big names looking to leave.
French forward Olivier Giroud is said to be considering a move away in a bid to book a place on the plane to Russia.
England manager Gareth Southgate has issued Wilshere with a similar ultimatum, saying he cannot get into the national team “without getting into the Arsenal team.”
There was talk that the midfielder could leave on loan, similarly to when he spent last season on loan at Bournemouth.
