Bernstein, who will leave the FA this summer, thinks England have struggled to transfer legendary players into ambassadorial roles in comparison with the likes of Germany and Franz Beckenbauer.
But following his work as an ambassador for London's 2012 Olympic and, most recently, Chinese football, former England captain Beckham could be well suited to a role with the FA when he quits playing at the end of the season.
Bernstein told Sky Sports: "From a Football Association point of point of view, I would pay tribute to him playing for England, the caps he has won, the pride he has shown and the passion he has shown.
"I think he sets a great example for young players with that passion of playing for England right across the game in this country.
"I wish him well, congratulate him on a fantastic career and, of course, I am sure the Football Association will be knocking on his door to see how he and the Football Association can work together."
Meanwhile, British Olympic Association Chair Lord Coe believes David Beckham should be the first man the Football Association call upon should they bid again to host the World Cup.
Coe worked closely with Beckham in the build-up to last year's London Olympic Games, and he insists the former England captain can have a major role to play in sporting politics.
Coe said: "I think he can do anything he wants to do. I would genuinely hope that if England ever really does attempt to put another proper bid together to win a World Cup that he is the first port of call, because he came into Olympic sport, understood it very quickly.
"Football is his sport and I can't think of anybody I would rather entrust to travel around the world on behalf of the English game than David Beckham.
"I can't imagine that David won't be in very high demand in all sorts of areas. I hope he gets used in the right and proper way because he is a very influential figure now in world sport."