Capello's contract expires after Euro 2012 but Bernstein would not be drawn on the timescale for finding a replacement - or whether he would rule out appointing another overseas boss.
In an exclusive interview with Sportsmail,former England manager Kevin Keegan warned leading homegrown candidate Harry Redknapp not to take up the post if it is offered to him, describing the job as 'soulless'.
Plenty to ponder: FA chairman David Bernstein has not ruled out a foreign successor to England manager Fabio Capello after Euro 2012
But Bernstein said: 'I cannot confirm that he will be English.
'I think there is a desire for the manager to be English but I wouldn't want to close the situation and we have to judge it at the time.
'I would not say at this stage that he has to be English, I would say at this stage it would be preferable.
'It is an important element, but not the only criteria, I clearly recognise there is a desire in this country to have an English manager when Fabio leaves.'
In the frame: Spurs boss Harry Redknapp is the leading homegrown candidate to replace Capello next year
Meanwhile, Bernstein admitted the controversial fixture scheduling that will see the FA Cup final and the likely Premier League title decider both take place on Saturday could be repeated next season.
Bernstein confirmed the Euro 2012 finals next year could mean top-flight league fixtures being played on the same day as the FA Cup final again - which is traditionally the last match of the season.
The final is also like to kick off at 5.15pm instead of the historic 3pm start.
Stoke manager Tony Pulis has criticised the scheduling which will see this season's FA Cup final between his side and Manchester City potentially overshadowed by Manchester United clinching the Premier League title at Blackburn just minutes earlier.
Bernstein said: 'The situation arises not because of the Premier League, but because of Champions League considerations and the fact the Champions League final takes place at Wembley in two weeks.
Stand off: Manchester City and Stoke prepare to contest the FA Cup final - but next year's showpiece could be moved from the traditional kick-off time of 3pm to 5.15pm
'We won't be able to have the game at the end of the season next year because of the European Championships and the need for the gap between the league and Euros.
'In the course of time, and it won't be next season - I can say that now unfortunately - we will certainly be looking to get the FA Cup on its traditional day.'
He added: 'We think we will be moving to a later FA Cup final kick-off, at 5.15pm which is potentially more popular with viewers.
'There was some reluctance in research to do that but all in all, it is a good move that will increase the TV value of the competition.'
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?Explore more:People: Tony Pulis, Kevin Keegan, Fabio Capello, Harry Redknapp Places: United Kingdom Organisations: Football Association
Source: Daily Mail
Source: Daily Mail