Tottenham manager Harry Redknapp has long been regarded as favourite for a role left open when Fabio Capello resigned in February in protest at the FA's decision to strip John Terry of the England captaincy.
But in a stunning development, the FA turned to Hodgson, who has a wealth of international experience having coached the likes of Switzerland and Finland.
FA chairman David Bernstein, thanking West Brom counterpart Jeremy Peace, 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 Hodgson, whose current Albion deal ends on June 30, has, unusually for an English manager, had spells with several overseas clubs, including Italian giants Inter Milan.
He took over at West Brom in 2011 after a brief and unsuccessful time in charge of Liverpool and the Midlands club are currently a respectable 10th place in the Premier League.
"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," Peace told West Brom's website.
"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 Premier League club.
"Everyone here has an excellent working relationship with him and he is immensely popular with our supporters."
It may be the FA felt able to speak to Hodgson at this time as the Baggies' season is effectively 'over', whereas they might have been accused of disrupting Tottenham, currently fourth and battling for a Champions League place, were they seen to speak to Redknapp before the end of the campaign.
And the fact Hodgson is shortly out of contract means the FA would not have to get involved in any expensive compensation deals, whereas prizing Redknapp away from White Hart Lane could prove extremely costly.
Redknapp has never managed outside of England whereas Hodgson, who has also been in charge of Blackburn and Fulham, started his coaching career in Sweden.
England are currently under the caretaker charge of Stuart Pearce, the former England defender who is also the manager of the national Under-21 side and coach of the British Olympic team.
Concerns have been raised about the FA's failure to bring in a permanent replacement for Capello so close to the start of the 2012 European Championships in Poland and Ukraine, where England begin the tournament against France in Donetsk on June 11.
England have two friendly internationals, away to Norway in Oslo on May 26 and against Belgium at Wembley on June 2, before Euro 2012 gets underway.