In fact, Klopp has told reporters "everything is fine" despite acknowledging Mane was "upset."
The Senegal international scored the Reds' second at Turf Moor to help send them above Manchester City and back to the top of the table. However, Mane made a passionate and animated exit when his number was called with five minutes to go.
Klopp was asked about the situation, but he preferred to play down its significance:
Neil Jones @neiljonesgoal
Klopp on Mane. "He was upset, that was obvious. Sadio cannot hide his emotions and I like that. But all sorted. We spoke about it and everything is fine. “We are individuals, we are emotional. It was a situation in the game he wasn't happy about. That's completely fine.' #LFC
Mane had been replaced by Divock Origi, but the source of his unhappiness might have been Mohamed Salah failing to tee him up for a tap-in shortly before the change:
If this is the root cause of Mane's frustration, it presents a potentially delicate situation for Klopp to deal with. Animosity between his two primary goalscorers could wreck the chemistry Liverpool have built and used as a platform for success in recent seasons.
The Reds have become a force again, both domestically and in European competition, thanks largely to the symmetry and production of a marquee front three. Mane and Salah are the star attractions ably supported by the vision and flair of unsung hero Roberto Firmino, who was also on the scoresheet against the Clarets.
Unlike nearest title rivals City, last season's UEFA Champions League winners are almost totally reliant on their front three for goals. It's likely why Klopp has couched his words in diplomacy when asked whether Mane could have a grievance against Salah.
There are those, including Uche Amako of the Daily Express, who believe Liverpool's No. 10 has every right to be angry with his team-mate:
Klopp appears content to laugh this brief episode off, but fostering some creative tension could actually work to Liverpool's advantage. This kind of environment would likely see Mane and Salah use an in-house rivalry to push each other to even greater heights.
They already bring out superb performances in each other:
Relying on tension to provide extra motivation for the strike partners to outdo one another might give the Reds the something extra they need to beat City to the title this season.
Ultimately, this question probably disappears long before it can be answered. Liverpool's forwards have supported each other excellently over the last few seasons without any drama.
Even so, it won't stop rivals from keeping a close eye on how Klopp's match-winners co-exist going forward during this campaign.