West Brom chairman Jeremy Peace on Sunday granted permission for the FA to hold talks with their 64-year-old manager, and Hodgson has already had a conversation with FA chairman David Bernstein.
No approach has been made to Tottenham boss Harry Redknapp, who had been viewed as the favourite to succeed Fabio Capello.
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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."
Hodgson is out of contract in the summer - meaning the FA would not have to pay any costly compensation or become involved in any wrangling with the club.
That is in contrast to the situation at Spurs who would expect hefty compensation were the FA to approach them about Redknapp.
Hodgson has coached the Switzerland and Finland national teams with considerable success in the past. He has also been extraordinarily successful on limited resources at West Brom, for whom he signed an 18-month contract last February after a disappointing 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 wants 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."