The Reds were linked with the 21-year-old over the January transfer window.
But a move did not transpire for the forward, who is rated as the second-most valuable player in the world at £223m.
But footage emerged on Saturday night of frustration getting the better of him in Ligue 1.
In the clip, PSG are taking on Montpellier at the Parc des Princes.
And the French champions are leading 5-0 with 22 minutes to go.
Mbappe has just been pulled off for Mauro Icardi – and the young forward is left fuming.
He can be seen arguing with Tuchel, who holds onto the 21-year-old to explain his decision.
But all the Frenchman wants to do is head to the bench.
After pushing away from the boss, Mbappe then shoves a coach offering a coat away.
This footage was shared on Twitter on Saturday night by popular fan show The Redman, where it has been viewed 2.7m times.
And the Anfield faithful made their feelings about the Frenchman clear.
“This would not fly with Hendo’s lads,” one supporter said.
Another commented: “I love Mbappe as a player but this s*** wouldn’t fly with Jurgen.
“Things are perfect – why disturb that?”
While a third gave Klopp some advice.
“Let him go to Real [Madrid],” they said.
“They’re used to dealing with prima donnas.”
And a fourth added: “He’s a fantastic player but I don’t want egos at our club.”
With a fifth simply writing: “No room for little brats.”
Liverpool do not have any problems upfront at the moment as Mohamed Salah scored two in their 4-0 win over Southampton.