The 25-year-old was stretchered off 34 minutes into the match, before being taken to a nearby hospital for treatment.
The club's head physio Kate Rees told reporters: "It's difficult at the moment to assess how long he will be out, because we need the swelling to go down first, but he is very unlikely to make the cup final."
A statement on the Swans website added: "Chico travelled with the rest of the squad to Dubai this morning for a well-deserved break in the sun.
"He will receive treatment from the club's medical staff in Dubai before a timescale for his comeback is confirmed."
Swansea will play Bradford City in the final in two weeks' time.