Wenger held talks with Stan Kroenke and Ivan Gazidis on Tuesday and the club are set to announce that the Frenchman will stay at the club until 2019.
Indeed, the Arsenal boss has agreed terms on a new two-year deal despite this season failing to clinch a top-four finish for the first time since he joined the club in 1996.
Arsenal beat Chelsea in Saturday's FA Cup Final to win a third trophy in four seasons, but the club's failure to challenge the top clubs for the Premier League crown has frustrated fans this term, leading to several protests against the manager.
And Lineker doubts Arsenal will break the top four next season, let alone maintain a title push.
The 56-year-old told the BBC: "Arsenal have got a long way to go at the moment. If you ask me now whether Arsenal are going to be in the top four next season, I would say most probably not. They are further behind than Wenger thinks."
Wenger has passed up the opportunity to leave Arsenal on a high by committing to the club for a further two seasons.
And Lineker believes it is "remarkable" that Wenger has been given the right to determine when he will leave the club.
He added: "The only other example of a manager that decides when and where he goes is Sir Alex Ferguson, but he was still winning league titles. Arsene Wenger was still winning FA Cups and until this season he had always got in the top four, but now that's changed."