The move has stunned the English game as almost everyone expected Tottenham manager Harry Redknapp would be top of their list.
With his contract at West Brom almost up, Hodgson would command barely any compensation, a factor which may just have helped the FA reach their decision.
FA chairman David Bernstein said: "Roy is the only manager we have approached and we remain on course to make an appointment within the timescale we set out soon after Fabio Capello's departure.
"Further conversations will now take place with Roy and my Club England colleagues before any further announcements can be made."
West Brom chairman Jeremy Peace said the club are currently in negotiations with Hodgson about his future.
But he added: "Roy has expressed a desire to explore this opportunity and, accordingly, the club have granted him permission to speak to the FA.
"Roy has done a fantastic job over the past 15 months and the fact the FA want to discuss the England role with him is testament to that.
"Roy is a proud Englishman and we can understand why he wants to speak to the FA about this highly-prestigious managerial position."
It is still possible Hodgson will say no but that seems unlikely as, in an extensive interview earlier this year, the Londoner said the job would be "a tremendous honour".
Hodgson has built up an impressive CV, which includes spells in the international game with Switzerland - whom he guided to the 1994 World Cup and Euro 96, although he resigned before the latter took place - Finland and the United Arab Emirates.
However the move has received a mixed response from England fans.
A prominent member of the England Supporters Club has demanded clarity from the FA over their decision to snub Redknapp.
Mark Perryman believes fans should be told why the Spurs boss appears to have been overlooked.
"I think first of all, the FA needs to be open in terms of the situation," he said.
"It's almost inconceivable they haven't approached Redknapp. I think the fans and the broader sporting public have a right to know what the situation is with Redknapp.
"It's quite conceivable he has turned the job down but fans have the right to know.
"I am not saying Hodgson is the wrong man but clearly there was a popular mood around Redknapp and in the spirit of open governance, let's be clear what the situation was there."
Perryman was impressed by Hodgson's experience and called for him to be given time to mould a team.
"Hodgson is the kind of guy who is willing to take the job on.
"He is a quiet and unassuming guy but the thing that impresses me is that when you look at his record he has managed all over Europe..."