Premier League referee Coote suspended over apparent video rant
It is unclear when the video was made or how it came into circulation and Britain's Press Association said it had not verified it.
According to the clip, Coote said Klopp was "arrogant" and used offensive language.
The BBC said the video appeared to refer to a match that Coote officiated between Liverpool and Burnley in July 2020, which finished 1-1.
Klopp criticised Coote after the match, saying the referee had been too lenient in his treatment of Burnley challenges.
Refereeing body Professional Game Match Officials Limited (PGMOL) said in a statement on Monday: "David Coote has been suspended with immediate effect pending a full investigation.
"PGMOL will be making no further comment until that process is complete."
Coote, 42, refereed Liverpool's 2-0 win against Aston Villa on Saturday.
Klopp left Liverpool at the end of last season after nearly nine years in charge at Anfield.
The German was critical of Coote over an incident in a match against Arsenal last season in which Gunners midfielder Martin Odegaard's hand made contact with the ball.
Coote, as VAR, did not advise on-field official Chris Kavanagh to review the incident.
Klopp said after the match: "How can a guy in an office see that and not come to the conclusion that maybe, possibly, it could be worth the referee having another look?"
Coote was also the VAR in the Merseyside derby at Goodison Park in October 2020 when Toffees goalkeeper Jordan Pickford inflicted a season-ending injury on Liverpool defender Virgil van Dijk.
Pickford was not sent off for the challenge and did not face retrospective action because, under Football Association rules, that can only happen if an incident was not seen either at the time or when it was reviewed by VAR.