Wilshere was told he could leave with around a month remaining in the summer transfer window but felt offers from other clubs were not enticing enough and he had yet to regain full fitness.
"It was an honest conversation," Wilshere told British media. "He (Wenger) said, 'I am going to be honest with you and at the moment we are not going to be offering you a contract, so if you can get a contract somewhere else, you can go'.
"Obviously, I was not happy with that but at the same time I was happy he was being honest.
"He also said at the same time that if I did stay I had an opportunity to fight for my place, and if I performed well in the League Cup and Europa League, I had a chance."
Wilshere has fought to regain his first team spot, making 31 appearances for the club this season and earning an England call-up for the first time since 2016.
The midfielder's contract expires at the end of the season and Wilshere is keen to seal his future before the World Cup finals begin in June.
"I have got three months left on my contract. Ideally, yes, I want it sorted as soon as possible. I want to go to the World Cup and enjoy it...," Wilshere added.
"Ideally from my point of view and the club's point of view, they will probably want it sorted."
England play a friendly away at the Netherlands on Friday before hosting Italy four days later.
(Reporting by Aditi Prakash in Bengaluru)
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