Premier League referee Clattenburg has been accused by Chelsea of using "inappropriate language" towards their midfielder John Obi Mikel during last weekend's match against Manchester United last weekend, an allegation he is believed to have denied.
In English top-flight football a referee, his assistants and the fourth official can hear each other via a microphone link but what is said is not recorded.
This contrasts with the situation in English Premiership rugby union, where a referee's remarks are taped and kept until the end of the season by the governing Rugby Football Union.
Moreover, for major club and international matches, spectators and broadcasters can listen in live to the referee's comments via the 'ref link' system.
Bernstein, while refusing to comment specifically on the Clattenburg case, said the issue of recording referees' on-field conversations should "definitely be looked at".
He also told BBC Radio Five's Sportsweek programme: "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.
"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 (football's world governing body) or IFAB (the International Football Association Board, responsible for the sport's rules) to be approved.
"(But) given what's happened recently, we should have an open mind about it and it should definitely be looked at."
Meanwhile Keith Hackett, former general manager of Premier League referees urged on-pitch conversations be recorded, telling the Observer: "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.
"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."
His view was supported by former Premier League and international referee Graham Poll, who said officials needed more support from governing bodies.
"The minimum that should happen is that (conversations are) taped," Poll told Sportsweek. "If an allegation comes up, we replay the transcript in private so the FA can go 'it's wrong, it's sorted, be quiet'."