Bernstein is holding talks with the Premier League in a bid to avoid a repeat of last season's situation, when Manchester City won the FA Cup on the same day as Manchester United won the title. The FA's understanding is that it did not work for the Premier League either and attempts are being made on both sides to afford the FA Cup a day to itself.
"Our ambition to have the FA Cup final back as the last game of the season won't happen in the immediate future," said Bernstein.
"But we are working very hard with the Premier League and other authorities to have the FA Cup final on a day to itself.
"We were very reassured with the last cup final. It happened on a day there were other Premier League matches and the title was decided - but there was no question it (the cup final) dominated the day.
"The credibility of the cup final was reinforced by that and hopefully everyone will see that and see it is the right thing for football that the cup final assumes that tradition of having a day to itself. The public want to see that."
Ultimately, the FA want to restore the cup final to its traditional place as the season finale - but that will not be possible for the foreseeable future.
The FA Cup final was moved earlier last season because Wembley hosted the Champions League final and the same will now happen in 2013.
The 2012 European Championships mean next season has to be completed by May 12 and 2014 is a World Cup year.
But the FA have been told that Premier League chief executive Richard Scudamore is exploring ways of juggling the fixtures, with the aim of keep Saturday clear for the cup final.