"The deal was closed this morning," Valencia president Manuel Llorente told Spanish television at the Champions League draw in Monaco.
"The player insisted he wanted to leave and to play in the Premier League. We wanted him to continue but in the end we have reached an agreement with Swansea."
The 27-year-old, who has been capped four times for Spain, worked with Swansea manager Michael Laudrup when the two were together at Getafe in the 2007-08 campaign.
Laudrup believes Scott Sinclair's move to Manchester City could still go through, despite the Welsh club rejecting a £6.2million bid from the Premier League champions.
The 23-year-old recently told Laudrup that he would not sign a new deal at the Liberty Stadium, and with less than a year to run on his current deal he was widely expected to leave before the transfer window closes on Friday.
"I can't say, I really don't know where we are at the moment," Laudrup said. "Even if talks have broken down there is still time for everything to happen, I am not saying he will go, or he will stay."
City were understood to have made an initial bid of around £5million, but Swansea were keen for a higher offer as Chelsea will get a percentage of any fee due to a sell-on clause inserted when Sinclair moved to south Wales in 2010.
City returned with an improved offer, but Press Association Sport understands it has been turned down by the Swans - who are determined to have their valuation of the player met. Laudrup would not confirm or deny that an offer had been rejected.
Laudrup is keen to bring in targets Pablo Hernandez and Marvin Emnes from Valencia and Middlesbrough respectively and the Dane expects there to be plenty of activity over the next few days.
"I think it will be very busy," he said. "Everyone here will have to be working hard over the next three days to make sure we are not in a situation where a club is saying 'the fax is not working'. I tried that, where a fax wasn't working in a club and they signed a player a little too late."
Laudrup was speaking after watching his side maintain their 100% start to the season by beating Barnsley 3-1 in the second round of the Capital One Cup at the Liberty Stadium, with Luke Moore weighing in with a double.
"I am always satisfied when we win," he said. "But I was not happy with the first 25 minutes and neither were the players."
He added: "I was pleased with the second half, there were moments after it went to 2-1 where we struggled for six or seven minutes, but more or less I was pleased with what I saw in the second half."