The defender, 21, fractured his leg against Burnley on December 5 and was at first expected to be out until mid-January.
But two months later he is still not back in training, dashing any hopes that he could make a comeback in February.
It is understood there are concerns from the medical staff at Melwood that the bone has not healed as it should.
Klopp was quizzed on whether the Reds star would need to go under the knife and he said: "I don’t know. We will see. It is possible, probably.
“It is not exactly going how we want, that’s how it is. He needs more time. We will see exactly how we do it.
“There were different moments when we thought we could do this or that. It will take time. We can not say more.”
It is not the news title-chasing Liverpool wanted as they look to win their maiden Premier League.
Gomez is Klopp's first-choice partner for Virgil van Dijk while Dejan Lovren has usually been the man to step in during the Englishman's absence.
But Croat Lovren, 29, is still struggling with a hamstring problem suffered in the FA Cup defeat at Wolves on January 7.
Although he made the bench against Leicester, he is not expected to be available for the trip to West Ham on Monday night, meaning Joel Matip looks set to retain his place.
With Gomez and Trent Alexander-Arnold still out and Nathaniel Clyne on loan at Bournemouth, versatile James Milner is likely to return at right-back at the London Stadium.
He missed the draw with the Foxes through suspension with Jordan Henderson filling in at full-back.
But Klopp is adamant he does not regret allowing Clyne to leave and now being short for cover.
He added: "I can imagine people think that but I don’t think about it.
“In the moment we did it there were different reasons. Then an hour later people could say ‘Why did you do that?’
“Milly got suspended and Trent injured or the other way around. Is that perfect? I don’t think so. You make the decision in the moment you have to and then people judge it later.
“Hendo did well there. We all know how good Clyney is, but would he have had more offensive influence in the game? I don’t know. Defensively, I did not see a mistake from Hendo.”
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