The Frenchman celebrates 20 years at Arsenal on 1 October and his contract expires at the end of this season.
The Football Association is searching for a new England boss following Sam Allardyce’s departure after 67 days in the wake of a newspaper investigation.
Allardyce left by mutual agreement after the Daily Telegraph claimed he offered advice on how to “get around” rules on player transfers.
England Under-21 boss Gareth Southgate takes charge of the senior team for the next four games as the FA assesses its options.
Southgate could be an option for the permanent role depending on his success, while Bournemouth boss Eddie Howe, Crystal Palace manager Alan Pardew and ex-Hull City manager Steve Bruce have also been mentioned as candidates.
Wenger is the longest-serving – and most successful – overseas manager in England, having won 15 trophies during his tenure at Arsenal.
His tally includes winning the Premier League three times and the FA Cup on six occasions.
BBC chief football writer Phil McNulty
The Arsenal manager’s credentials easily outstrip those of the other candidates being touted as Allardyce’s replacement.
At 66, Wenger might even share Allardyce’s view when he was appointed that his age and experience make him the perfect fit for international management.
Wenger is the perfect next England manager with the ideal credentials and track record if the FA can formulate a plan to somehow attract him to what many now call an impossible job.