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."
Following the friendly against the Dutch, England play Norway on May 26 and Belgium on June 2 before the Euros where they face France on June 11, Sweden on June 15 and Ukraine on June 19.
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.
PEARCE LONG-TERM?
"I've got a big job to do, I've got a big game on Saturday with Tottenham so Tottenham is my focus."
Tottenham are also unlikely to allow Redknapp to leave before the end of the season at the earliest as they continue their push for a probable Champions League spot next season while retaining an outside chance of landing their first League title since 1961.
They are currently third in the Premier League, seven points behind leaders Manchester City with 14 matches still to play.
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.
However, FA General Secretary Horne said they were fully aware of the widespread support for Redknapp.
"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.
