Vice-chairman David Gill on Friday admitted his organisation were set to bid for the semi-finals and final of the tournament, which will be played in 12 different countries.
And Bernstein, who is this summer set to leave the FA, has affirmed the governing body's desire to welcome a major competition having painfully missed out on the 2018 World Cup.
Wembley will on Saturday evening host the UEFA Champions League final for the second time in three years and Bernstein is confident it is another opportunity to showcase the country.
Speaking from Wembley, Bernstein told Sky Sports: "We have got wonderful facilities here.
"I believe our World Cup bid, technically, was the best of the whole lot. We did not get it for other reasons.
"But, of course, we can hold a tournament here. We will be bidding for part of the 2020 European Championship. I know that for sure.
"So the quicker we get something here [the better]. The public wants it, the FA wants it, I want it.
"It is something I believe we deserve. It is a long time since we last had a World Cup here."