The Spanish centre-back has formed a fine partnership with Ashley Williams at the heart of the defence since signing from Genoa in the summer but he was stretchered off after 34 minutes on Saturday.
The 25-year-old needed prolonged treatment after making a stooping header and was then taken to hospital, where scans revealed the full extent of the injury.
And, with the Wembley final against Bradford in just two weeks' time, on Sunday 24th February, Flores will probably not have enough time to recover.
Swansea's head physio Kate Rees said on Sunday morning: "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 Swansea's official 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."
If Flores is on the sidelines for the final, Swansea boss Michael Laudrup will have to decide how to reconfigure his defence.
Kyle Bartley came off the bench to replace Flores on Saturday and club captain Garry Monk is another option for Laudrup.