The duo were seen arguing when Salah was about to come on against West Ham United in a 2-2 draw.
According to Sportbible via lip reader John Cassidy, Salah said: There's nothing I can do about that, nothing."
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Klopp's response is not thought to be one that the lip reader could easily make out.
Salah then said: I will, I will get a red card, after seven years, after seven years' service, after seven years!"
The Egyptian did walk by reporters after the game, saying If I speak, there will be fire."