It follows an approach from FA chairman David Bernstein to West Brom chairman Jeremy Peace, which was subsequently accepted.
No approach has been made to Tottenham boss Harry Redknapp, who had been viewed as the overwhelming favourite to succeed Fabio Capello.
David Bernstein said: "I'm grateful to Jeremy and all at West Bromwich Albion for their co-operation in allowing us to approach Roy, who I have since spoken with.
"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."
The 64-year-old has been repeatedly linked to the role, despite the fact that Tottenham boss Redknapp had been regarded as the most likely candidate to be appointed as Capello's successor.
Hodgson has previously said: "It would be an honour of course if ever the offer came my way, because it's the pinnacle of success in a way and certainly of coaching success if you are ever invited to manage your own country."
Asked if he would take the job if offered it, Hodgson said: "Of course, I think that anyone would."
Hodgson has coached the Switzerland and Finland national teams with considerable success in the past, and also managed the United Arab Emirates side.
He has been successful on limited resources at West Brom, for whom he signed an 18-month contract last February after a disastrous spell at Liverpool.
West Brom said in a statement: "The club are currently in negotiations with Roy about a new contract - his current deal ends on June 30 - and remain very keen to bring these talks to a successful conclusion.
"However, Roy has expressed a desire to explore this opportunity and, accordingly, the club have granted him permission to speak to the FA."
Peace added: "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.
"However, we have emphasised to Roy how much we would like him to remain as our head coach and continue his major contribution to our project at The Hawthorns as we look to establish ourselves as a Barclays Premier League club.
"Everyone here has an excellent working relationship with him and he is immensely popular with our supporters."