The Frenchman was carried off on a stretcher after going down in agony following a challenge on Adam Webster late in the first half of the Premier League clash at the Etihad Stadium.
City have already lost winger Leroy Sane, to a long-term knee problem this season and the club are anxious Laporte has not suffered the same fate.
City manager Pep Guardiola, who described Laporte as the world's best left-sided centre-back only on Friday, said: "I think he is in hospital. It doesn't look good but I don't know exactly what he has done. Tomorrow, or tonight, we will know exactly what he has.
"He didn't feel good. The doctor is going to call me and tell me what he has but I think, for a while, he will be out. Hopefully it is not as long as Leroy but we will see."
Laporte's injury seems set to deprive the 25-year-old of the chance to make his France debut in his country's upcoming Euro 2020 qualifiers against Albania and Andorra.
With John Stones still sidelined, Laporte's exit also forced midfielder Fernandinho into emergency action as a centre-back, highlighting City's shortcomings in the area after Vincent Kompany's departure.
Guardiola, however, has no doubts Fernandinho can fill the position.
He said: "He is a holding midfielder, everyone knows it, but I think he has the conditions to play there.
"The problems we have right now, we have a situation, and he is going to adapt. He is going to help us because he is intelligent, fast, strong in the headers, in the build-up. He has the quality to see what happens in front. He can play in both positions."
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