The FA has been criticised for not handing a role to Moore, who died 20 years ago today.
In a column for the Sunday Times, Bernstein conceded English football's governing body should have made more of 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, who won 108 caps for his country and went on to manage Oxford City and Southend, was awarded the OBE but never offered a post within the FA.
"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."