The right-back, 28, was hauled off at the interval following his role in Liverpool's opener, which came right on the stroke of half-time.
Aurier was caught napping as Sadio Mane ran behind him and squared for Roberto Firmino to score Liverpool's first Premier League goal of 2021.
Trent Alexander-Arnold doubled the lead almost immediately after the break, before Spurs hit back through Pierre-Emile Hojbjerg, but Mane made sure of Liverpool's victory later in the second-half.
The Telegraph reports that Aurier didn't stick around to watch the dramatic second-half, having stormed out of the stadium after being substituted at half-time.
Meanwhile, the Daily Mail claims it 'kicked off' in the home dressing room at half-time due to fury surrounding the goal which put Liverpool in front.
When asked about his team-talk, Mourinho said after the game: "It's the mood of a team that it was difficult to accept you are losing, difficult to accept the nature of the goal because the goal is in some aspects a replica of the chance they had in the first minute.
"So it's of course a mood where people are not happy - but then we have to move. Then Harry is injured and we need to try to do something. Unlucky that we started the second half conceding a goal."
Harry Kane was also pulled at half-time due to an ankle injury, sparking fears for of another long-term lay-off Tottenham and England.
The Spurs striker will undergo scans in the next 48 hours and Mourinho admitted the initial signs are not good.
He said: "Two ankles, the first was a bad tackle, it was Thiago, the second one I didn't know well.
"But two injuries in both ankles, the second one worse than the first one - a few weeks, I don't know.
"There are some players you can't replace. When it happens it happens but I believe we have to fight against it, we cannot do anything else."
Serge Aurier reportedly left the Tottenham Hotspur Stadium at half-time of Spurs' 3-1 defeat to Liverpool after a heated inquest by Jose Mourinho.