Michael Laudrup revealed, as expected, that Chico Flores' desperate bid to prove his fitness had fallen short.
The Spaniard was rated "very unlikely" to play any part after rupturing ankle ligaments against QPR.
The 25-year-old threw himself into his rehabilitation in an effort to play some part, but Laudrup will instead choose between Kyle Bartley and club captain Garry Monk to partner Ashley Williams.
"Chico tried today to go out and train, but it was much too early, we all knew that," said Laudrup.
"It is an injury you would normally estimate to be out for four to eight weeks and this Saturday would be two weeks (since the injury).
"It is such an important game but today he realised it is much too soon and he has to accept the fact."
Laudrup also confirmed Gerhard Tremmel will start in goal at Wembley ahead of Michel Vorm, having started in all previous rounds of the League Cup.
There would have been astronomical odds on the Swans and Bantams making it to Wembley, and even at the semi-final stage a Chelsea against Aston Villa final looked a far more likely line-up for Sunday's showpiece.
Even though it will be the first major cup final in Swansea's history, the Welsh club are well aware the majority of neutrals will be firmly rooting for their League Two opponents, who are the first side from the fourth tier to reach this stage of the competition since Rochdale in 1962.
That means a change of mindset is necessary for a side who have relished the tag of underdogs since securing promotion to the top flight in 2011.
Laudrup said: "I would think if you go back, it's very rare to see a major final between a small team in the top flight and a team from the lower leagues.
"When teams from lower down get to the final it is normally against Manchester United, Manchester City, Chelsea, Arsenal, Tottenham or Liverpool - like when Birmingham played Arsenal.
"This is different because although we are one of the lower teams, we have to accept we are favourites and we must take the initiative in the game.
"We have no problem with that and we accept the majority of people will want Bradford to win as they are the underdogs.
"We have seen what they have done to Wigan, Arsenal and especially to Aston Villa over two legs.
"It is fantastic, if you say what we have done is a fairytale, then the fact Bradford are there is a huge fairytale."