Odds on the Frenchman taking charge of the national team have shortened drastically since Sam Allardyce's 67-day reign ended on Tuesday.
Wenger has been Arsenal manager for 20 years but his contract is up at the end of the season.
"If he feels the situation is right at Arsenal and gets offered a new contract at Arsenal and he and the board are together with the way the club is going he will re-sign," Winterburn told Reuters.
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"But if he doesn't, because of his style of play I would love to see him be the England manager. I would love to see him change the attitude of the English players and the public because right now we are at a very low ebb.
"People are fed up with it. Personally I want to see him stay at Arsenal but he would be very good for the England team."
Given the choice between remaining at Arsenal and taking the England manager's job, Winterburn believes Wenger would choose the former, although he said the FA should try to persuade him.
"Whoever comes in needs to do something radical because we just don't perform in tournaments," Winterburn, who spent 13 years at Arsenal, the last four under Wenger when he earned a Premier League and FA Cup double in 1998, said.
"But if Arsenal offer Arsene a new contract he stays, full stop."
Allardyce left the England job for "inappropriate" conduct following secret filming that showed him offering advice to businessmen on how to "get around" rules on player transfers.
(Reporting by Martyn Herman, editing by Pritha Sarkar)
