On Monday afternoon, a video was leaked onto social media which appears to show Coote calling Klopp a "German c***" and "f****** arrogant" at some point during a COVID-19 lockdown a few years ago.
Within a couple of hours of being notified of the video, which has now been viewed over 50 million times on X - formerly known as Twitter - Coote was suspended by the PGMOL, who are now investigating whether the footage is real.
The official has supposedly accepted that the video is genuine, albeit while not being able to recall the discussion, and Watts could envisage his fellow referees letting out "a collective groan of disbelief" when the footage was posted.
"I just don't know. Well, I was gonna say, I don't know what he was thinking. I'm not sure he was thinking, judging by how he came across in the video," Watts told Sports Mole.
"I think he was definitely a bit of an after party at some point. It had clearly been a long night. And to put yourself in that position as a top Premier League referee, just cannot have been thinking straight at all.
'Coote video is a dreadful look for PGMOL'
"And obviously, it's come back to bite him now a few years further down the line. I mean, it's a dreadful, dreadful look for PGMOL at a time when everyone, you know, the focus is on them week after week. Anyway, I can imagine Coote's colleagues, just the sort of collective groan of disbelief when they all saw the videos like this is going to make our job even harder."
Following the bombshell leak, Liverpool fans naturally began to trawl through the archives to find instances where Coote gave a controversial decision against their club, and the 42-year-old has been involved in four high-profile contentious calls against the Reds.
One of those happened as recently as Saturday, as he refereed Liverpool's 2-0 Premier League win over Aston Villa, where he waved away shouts for an apparent last-man foul on Mohamed Salah by Leon Bailey before Darwin Nunez opened the scoring.
Coote was also on VAR duty for last season's Liverpool-Arsenal match at Anfield, where he dismissed appeals for a Martin Odegaard handball inside the Gunners box, and he was in Stockley Park for the Merseyside derby at Goodison Park in October 2020.
During that 2-2 draw, Liverpool captain Virgil van Dijk suffered his season-ending ACL problem in a tackle by Jordan Pickford, who was not punished for the incident, and a last-gasp Jordan Henderson winner was also ruled out for a contentious offside.
Watts has shot down baseless rumours that corruption is present among officials, but he has conceded that he finds it "very hard" to believe that any personal bias does not "creep in".
''It's hard to think personal bias does not creep in'
"There's always that feeling, certainly amongst fans, of bias and corruption and stuff like that. And I've never bought into that. I've never believed that corruption or anything like that has anything to do with this," Watts said.
"But bias, I think, as a human being, it's very hard, I would suggest, because these are all big football fans. You don't get into refereeing unless you're a big football fan and you love the sport. So you're clearly going to have an allegiance to a club, whether that disclosed or not.
"And so it's very hard to think that maybe personal bias doesn't sometimes creep into something. You would hope it doesn't. But human nature just dictates that it might.
"Fans will always be thinking that in the back of their minds and just seeing something like this. If you're a Liverpool fan now, you're obviously going back over every decision that David Coote has taken against your club and saying, well, it makes sense now and things like that. And I'm not saying that he has made those decisions because of bias, but that's just the feeling when you watch a video like this.
"That's the feeling as a fan you're going to think. So it's a terrible look for PGMOL and a terrible look, obviously, for David Coote. An investigation is now going on. We'll have to wait and see how what the sort of findings are from that.
"But it's tough to see a way back for him after something like this. And, yeah, it's come at the worst possible time for Howard Webb and PGMOL, given how much scrutiny that organisation is already under."
Liverpool expert David Lynch has also had his say on the matter regarding the alleged video of Coote, who is now the subject of an FA investigation as well as the PGMOL.