The FA said the vastly-experienced Hodgson, 64, was the only candidate they had approached about replacing Italian Capello who quit in February following a row over deposed England captain John Terry.
Tottenham Hotspur manager Harry Redknapp had been widely considered to be the strong favourite to get the job. Redknapp has repeatedly said he is totally committed to Spurs and would not talk about England unless he was offered the position.
"West Bromwich Albion have today granted permission for the FA to speak with Roy Hodgson regarding the position of England manager," the FA said in a statement.
FA chairman David Bernstein added: "I'm grateful to [WBA chairman] Jeremy [Peace] 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."
England meet Belgium in a friendly on June 2 before opening their Euro 2012 campaign against France in Donetsk on June 11. Sweden and co-hosts Ukraine are also in England's group.
"Roy is a proud Englishman and we can understand why he wants to speak to the FA about this highly prestigious managerial position," Peace told the club's website.
"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."
