During the meeting it became clear Capello felt he could not keep his position because of the decision to remove John Terry as captain.
The team is now without a manager and captain, just four months before Euro 2012 begins.
Capello said: “I leave England and I will not say anything else.”
Bernstein had excluded Capello from discussions over Terry’s role.
It is rumoured that the FA will ask under-21 coach Stuart Pearce to step up as a caretaker while a replacement is found.
Harry Redknap is already a firm favourite for some, with Wayne Rooney tweeting: “Gutted Capello has quit. Good guy and top coach. Got to be English to replace him. Harry Redknapp for me.”
But this riled Old Trafford captain Roy Keane, who said Rooney should “keep his nose out of it”.
He added: “He didn’t do enough on the pitch when Capello was manager.
“Wayne should concentrate on being a player and not on who he wants as England manager.”
Sunderland manager Martin O’Neill said: “Harry is obviously the favourite and I think he should be. He deserves the opportunity, he should be the one.”
Prime Minister David Cameron said: “England now needs a new coach and a new captain and I hope we can get on with that and make the best of the opportunity this summer.
“I am sorry to see Fabio go. I think he was a good coach and a good man. I don’t think he was right about the John Terry issue. You can’t be captain with that question mark that needs to be answered.”
But asked if Redknapp should be the next manager, Cameron said: “The day when the Prime Minister picks the England coach will be a very bad day for football but I am sure we will find someone really good and I am sure that we will play well when the time comes.”