Gomez sustained the injury during a training session with England yesterday and was withdrawn from the squad before this morning's friendly against Ireland.
Liverpool said in a statement that Gomez had undergone surgery to repair a tendon in his left knee, with "no damage to any other associated knee ligaments".
"The operation to correct the damage, which took place in London on Thursday morning, was a success and the 23-year-old will make a full recovery," the club added.
"No time scale is being placed upon his return, though the issue is likely to rule him out for a significant part of the remainder of 2020-21."
Gomez's injury is a massive blow to manager Jurgen Klopp whose side are already without key centre back Virgil van Dijk because of a season-ending knee injury.