The 18-year-old was carried off on a stretcher during the 3-0 win at Leeds on Sunday with the serious injury and had an operation on Tuesday.
“I spoke to him the night after the game and he was in the best possible place at that moment,” said Klopp.
“Today he is in London and will have surgery today. We all wait for news from that.”
Elliott’s team-mates were clearly disturbed by the seriousness of the injury at the time but they have all rallied round since.
“What happened with Harvey made me sad. It’s difficult to deal with that when you’re on the pitch,” said goalkeeper Alisson Becker.
“All the players have a big role to support him, we need to do our best in this time.”