The FA feel the Gunners boss is an ideal man to take over from Hodgson after the humiliation of losing 2-1 to Iceland on Monday.
Wenger’s existing deal still has a year to run, and there is a hope within Wembley they may be able to tempt the Frenchman with the opportunity to finally manage at international level.
England have failed with an approach before, but at that time Wenger was locked into a long-term commitment to Arsenal that helped secure the finances for the development of their new ground.
At 67, Wenger is in a different position – although he has always maintained he would never walk out on a contract during his 19-years with the London club.
But the Gunners have no intention of seeing their manager leave at all, and are even preparing to offer him a new two-year deal to remain at the club.
They remain confident he will not have his head turned by the England job, and there is little prospect any job-share deal would be entertained either at Wembley or at Arsenal.
The FA have been caught out by Hodgson’s sudden departure, even though there was always the prospect he would be leaving this summer.
There are no obvious candidates among English club managers while the prospect of under-21 boss Gareth Southgate stepping up is one that hardly excites.
But England bosses are in such a mess they are now facing the prospect of handing the reigns to Southgate on a short-term deal while they look around for other alternatives.