The four-man panel which will decide who takes over from Capello are Bernstein, FA general secretary Alex Horne, FA director of football development Sir Trevor Brooking and managing director of Club England Adrian Bevington.
Bernstein approached West Brom on Saturday and was given permission to speak to Hodgson, who is likely to be appointed on a long-term basis.
It is expected any contract will cover the three tournaments up to and including Euro 2016, at which point he will be two months short of his 69th birthday.
Hodgson's contract with West Brom ends on 30 June.
Tottenham boss Harry Redknapp had been heavily linked with the job ever since
Capello left the post in February,
but the FA has decided to discuss the position with Hodgson.
It is understood the FA's main concern about Redknapp was whether he shared their vision for the
National Football Centre in Burton,
designed as a £100m centre of excellence for coaching.
The FA felt Hodgson would be more willing to lead from the front on that issue.
On Friday, BBC Radio 5 Live's Pat Murphy spoke to Hodgson about the England job, and put it to the West Brom manager that the FA was expected to make a move for Capello's successor this week.
"It's news to me," Hodgson said. "I shall just watch like everyone else interested in football to see who the next England manager will be.
"I don't what the plans are of the FA, as I've said all along. They've got a job to do. I'm sure they have looked into the sort of candidates that they consider potentially right for the job.
Motty on Hodgson's prospects
"No doubt, if you say next week, we'll find out which candidate they have plumped for."
Murphy told Hodgson he "ticked a lot of the boxes", and the West Brom manager replied: "I don't know about that. It's nice of you to say so, but I'm sure there's lots of other people who tick a lot boxes as well."
He has been in charge of 18 teams either at domestic or international level, including being the manager at Inter Milan where he was also technical director in a second spell at the San Siro.
When the FA made its approach, Albion chairman Jeremy Peace had been hopeful that Hodgson would stay at The Hawthorns.
Peace said: "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.
Ian DennisBBC Radio 5 live's senior football reporter
"This is an appointment for the long term with the FA looking at the wider picture. It's four weeks until the penultimate warm-up match for Euro 2012 but this decision is based on benefiting English football for the next four years.
"The choice of the next England manager was always going to divide opinion but nobody can argue with the level of experience Hodgson possesses at international level. That will be invaluable as he looks to harness the young talent and embrace the new home of English football development, St George's Park, to lay down a structure for the years ahead."
"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."
England have two fixtures before their Euro 2012 opener against France on 11 June, with friendlies planned against Norway on 26 May and Belgium on 2 June.
Stuart Pearce - who is in charge of both the England Under-21 squad and the Team GB Olympic set-up - is currently in temporary control of the national side and oversaw a
3-2 friendly defeat by the Netherlands
at Wembley last month.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.