Clattenburg has been accused by the Blues of using "inappropriate language" towards John Obi Mikel during the team's 3-2 defeat against Manchester United at Stamford Bridge last week and Bernstein remains mindful of the growing clamour for change.
"The referee is miked up with the assistant referees and the fourth official so they are all aware of the comments and the dialogue that takes place, but it's not recorded," Bernstein told BBC Radio Five's 'Sportsweek'.
"In order for that to happen, the referees' organisation – the Professional Game Match Officials Board – which is controlled by the FA, Premier League and Football League, would have to recommend that way forward.
"Then it would need to go before Fifa or IFAB [International Football Association Board] to be approved."Given what's happened recently, we should have an open mind about it and it should definitely be looked at."
Former general manager of Premier League referees, Keith Hackett, believes a move towards recording match officials can be seen as a positive move forward for the game.
"The FA are members of IFAB, so could bring forward a proposal to permit the recording of the conversations between players and officials during a game," Hackett told The Observer.
"I would have no problem with fans having the ability to purchase a 'Ref Link' to listen in to what the match officials are saying."And given what has happened in the last few days and, with no knee-jerk reaction, I would like to conversations between the match officials for the full duration of the game recorded."