Salah and Klopp were seen arguing on the sideline as the Egyptian was being brought on against West Ham.
The Reds could only draw the game 2-2, dealing a death blow to their title contention.
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Per The Sun, Klopp was urging Salah to shake his hand before going onto the field.
A source told the outlet: "This has happened occasionally with Jurgen and other players.
"He is very big on interaction with his subs before they go on and felt it didn't happen at first with Mo.
"Mo was obviously annoyed at being left out of the team and wasn't in the best of moods while Jurgen wasn't happy with his behavior."