Michel Platini, the Uefa president, has previously suggested the tournament could be staged in multiple different countries in the continent in eight years' time, with further discussions set to take place in December.
And FA chairman Bernstein has suggested Wembley may be put forward as a venue should the plan come to fruition.
"Clearly Wembley is incredibly highly thought of by Uefa and it is something we will probably push for," Bernstein said at a media briefing.
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"Uefa want to hold the semi-finals and the final on the same ground, or in the same city and I think we would be on their shortlist - but there would be some strong competition."
The FA was unsuccessful in its bid to host the 2018 World Cup, and while Bernstein stated he would not wish to be drawn into a similar bidding process, England would be keen to stage Euro 2020, though Turkey remain favourites if they are not awarded the Olympic Games during the same summer.
Bernstein added: "One of the major factors is whether Turkey get the Olympic Games or not. If Turkey do not get the Olympic Games then I think they are in a pretty good position to stage the Euros.
"If we believed there was a real opportunity of having the tournament here of course we would look at it.
"Of course there would be an intent, the public want it and we'd want it and it would be wonderful to have it here.
"If Michel Platini's current views prevail and it becomes a pan-European competition, then the focus needs to be on the semi-finals and final."
He continued: "We would not want to be involved in a World Cup-style bidding process.
"We do not want to spend a lot of money on something that fell flat."