The FA will meet on Friday to begin their search for a new manager following the sudden resignation of Italian Fabio Capello on Wednesday, with Tottenham Hotspur's Harry Redknapp the hot favourite.
Stuart Pearce, England's Under-21 manager who is also boss of the London 2012 British Olympic squad, will take charge for the friendly against the Netherlands at Wembley on February 29 but Bernstein said after that they had a totally "open mind" about the future.
When asked at a news conference if England could go to the tournament with a caretaker coach, Bernstein replied: "I would say that is a possibility, all options are open.
"We are not prepared to restrict ourselves at this stage. He might be English, he might be British, he might not be, he might be for the Euros only, he might be long-term.
"We need to look at all the options. It would be pointless, 24 hours or less after we have lost our manager, to start limiting ourselves."
While some England players and almost every pundit have called for Redknapp to be installed as Capello's replacement, the 64-year-old said on Thursday that he had not thought about managing England and was totally focused on his job at Spurs and their next match on Saturday against Newcastle United.
Speaking a day after being found not guilty on two charges of evading tax after a 13-day trial in London, Redknapp told reporters: "I don't know anything about the England job. I've not thought about it.
"I've got a big job to do, I've got a big game on Saturday with Tottenham so Tottenham is my focus."
Bernstein, and fellow FA officials Alex Horne, Adrian Bevington and Trevor Brooking, would not be drawn on Redknapp's possible appointment on either a short-term or long-term basis.
"Of course we understand the feeling of the nation and the supporters. The England fans are very important when it comes to choosing the right person," he said.
"But we have to sit down and look long-term and give ourselves as many options as are available. We owe it to ourselves to write-up the job brief and we've got an exhaustive list."
Bernstein added that the FA's immediate concern had been to get a manager in place for the Netherlands game, explaining: "We have got a match in three weeks' time. We need a short-term quick answer.
