Kompany, who had not played since damaging his thigh in a Champions League semi-final against Real Madrid on May 4 and has experienced an injury-hit City career, departed during the dying seconds of the EFL Cup tie.
Goals from Gael Clichy and Aleix Garcia ultimately sealed the deal for City but substitute Gylfi Sigurdsson pulled one back for Swansea in injury time, after which Kompany departed down the players' tunnel as the home side pushed in vain for an equaliser via a last-gasp corner.
City manager Pep Guardiola said: "The doctors are speaking with him and seeing what's up. Hopefully, it will not be a big problem
We're going to wait for the results from the doctors."
Asked if he had spoken with Kompany, Guardiola added: "No, not yet.
"(His performance) was good
It's not easy after a long, long time without playing
Defensively, he was so strong - he played really good.
"Once they scored the goal, after I said: 'What happened?' They said that Vincent had gone to the locker room
Hopefully, it will not be a big problem
We will have to wait.
"If he's injured, he's injured
He cannot stay on the pitch."
City recorded their ninth successive win under Guardiola, seeing off Swansea 2-1 despite starting with a team that showed nine changes following last weekend's 4-0 win against Bournemouth.
Swansea, who have now gone four league and cup games without a victory, had their moments in an entertaining encounter but the visitors had an extra touch of class about them.
They were also helped by second-half substitute Kevin De Bruyne's introduction as they moved into the last 16 three days before facing Swansea again, when Premier League points will be at stake.
"Garcia played good, and I want to congratulate the young players tonight
They played good," Guardiola said. "Fernando made an amazing effort, because he had been injured for the last two weeks
He played a fantastic game."
Swansea manager Francesco Guidolin, meanwhile, praised his players after they delivered a vastly improved display following last Sunday's tame 1-0 defeat against Southampton.
"I am disappointed because I think we deserved another result," he said. "We deserved to draw and take it to extra time
The game was well-balanced and the performance of my team was good.
"It is not a good moment for us
Little things are going against us, we are missing chances. Manchester City are a very good team, even though they changed their players around
But I am optimistic, as I saw my team play a good game."
Asked whether he was feeling any pressure after the Swans' recent run of results, he added: "It has not affected me.
"This is my job
With my experience, I know football is like this
it does not bring any added pressure for me
I have to work with my players, my staff and to pick up better results."
Source : PA
Source: PA