The Spanish centre-half, who has formed an excellent defensive partnership with Ashley Williams since arriving from Genoa in the summer, was stretchered off after 34 minutes of the 4-1 win over the Rs.
Flores went down after stooping to make a low defensive header, and required prolonged treatment. He was taken straight to hospital where a scan revealed the full extent of the damage.
"It's difficult at the moment to assess how long he will be out because we need the swelling to go down first," said Swansea's head physio Kate Rees.
"But he is very unlikely to make the cup final."
Flores has travelled with the rest of the Swansea squad to their warm-weather training camp in Dubai, where he will begin his rehabilitation.
But it is a blow for manager Michael Laudrup with just two weeks before the Wembley meeting with npower League Two Bradford.
Kyle Bartley replaced Flores against QPR, while club captain Garry Monk is another defensive option for the Dane.