The FA has never previously stated in public its regret over its treatment of Moore, but writing in a column for the Sunday Times 20 years to the day since Moore died from cancer, Bernstein said the game's national governing body should have utilised the former West Ham defender's expertise.
"Bobby was the man who led England to our ultimate moment of football glory. He remains an eternal credit to his family, friends and everyone involved with West Ham United," Bernstein wrote.
"I am aware the Football Association has been criticised over its treatment of Bobby once he retired from football.
"It saddened me that this is the case and while I am not privy to exactly to what happened at the time, it is clear to me the organisation could have done more."
Moore won 108 caps during his England career, the highlight being leading the national side to victory over West Germany in the 1966 World Cup final at Wembley.
"If Bobby were alive today I am sure we would have asked him to be the chief ambassador for the Football Association in its 150th year," Bernstein said. "He was simply one of the nation's greatest-ever footballers."
Moore was awarded an OBE but was never given a role within the FA, although he did stay in the game after he retired, taking up managerial roles at Eastern AA in Hong Kong, Oxford City and Southend.