A video which began circulating online on Monday appears to show Coote making highly derogatory remarks about Klopp, including reference to the German's nationality, and the Anfield club.
The FA has the power to investigate remarks made in a private context and it is understood part of its investigation will centre on whether Coote's remark about Klopp's nationality constitutes an aggravated breach of its misconduct rules.
Coote was suspended by Professional Game Match Officials Ltd on Monday pending a full investigation.
"We became aware of a video which was being circulated on social media that we are taking very seriously," PGMOL chief Howard Webb told Match Officials Mic'd Up on Sky Sports.
"We have instigated a full investigation and one of our referees, David Coote, has been suspended with immediate effect pending the outcome of that investigation. There isn't much more I can say pending the outcome of that investigation."
In the video, Coote is asked for his views on a Liverpool match where he has just been fourth official, and describes them as "s***".
He then uses an offensive term to describe Klopp and, asked why he felt that way, Coote says the German had "a right pop at me when I reffed them against Burnley in lockdown" and had accused him of lying.
"I have got no interest in speaking to someone who's f****** arrogant, so I do my best not to speak to him," Coote says.
Klopp left Liverpool in the summer after almost a decade in charge. Coote was the official for the Reds' most recent Premier League game - a 2-0 win over Aston Villa on Saturday.
Retired referee turned pundit Mike Dean believes the controversy will have a damaging effect on officials at every level of the English football pyramid.
"He'll be devastated, it's damaging for everyone involved" 💬 — Sky Sports Premier League (@SkySportsPL) Former referee Mike Dean shares his thoughts after David Coote was suspended by the PGMOL yesterday.pic.twitter.com/1Q9eBn95jwNovember 12, 2024
"It's damaging for everybody involved. When I saw it yesterday I couldn't believe it," he told Sky Sports.
"As referees you can't put yourself in that position no matter what you do. You can say things away from camera, you can talk amongst your friends and things and colleagues, but you can't let somebody video this and then hope that it's never going to come out."
Dean still believes Coote could return to refereeing if he wanted to do so - though admitted the 42-year-old would face an uphill battle.
"I don't think he's damaged his career beyond repair. I'd like to think that they (PGMOL) can come to some kind of agreement," added Dean, who has had a text message exchange with Coote but would not reveal its nature.Â
"Would David still want to carry on refereeing? Obviously it will be hard to come back in and start again."
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