The winger, who earlier this year switched allegiance from Portugal to the nation of his birth Angola, made a late deadline day loan switch from the Whites to the Bats.
Valencia's interest was registered well before the final moments of the transfer window, but Costa's departure was very much dependent on Leeds being able to bring in Dan James from Manchester United. Once that £25m deal was complete, Costa was always likely to leave Elland Road, with Sheffield United, Turkish clubs and a pair of La Liga outfits all interested. Valencia won the race and although three years remain on his Leeds contract, Costa is not expected to feature again for the Whites.
A loan deal, with an option for Valencia to make it permanent, was nearing completion three hours before the 11pm deadline, but confirmation that the paperwork had been successfully registered in Spain did not come until after midnight.
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It was not a difficult decision, Costa told Valencia's in-house media team, thanks in part to information he gleaned from Rodrigo, who swapped the Mestalla for Elland Road last summer in a £27m deal, and Gonçalo Guedes, who was his team-mate at Benfica.
"The first thing I did was talk to Rodrigo Moreno," said Costa.
"He told me many good things both about the club and the city and it was an easy decision. I am happy to be here. Guedes was my partner at Benfica. I've known him for many years. I spoke with him before coming and he told me that the Club had all the conditions for me to continue evolving.
"Valencia CF is a very big team. I'm very excited and I want to start working. I want to do my best for the team.
The other conversation Costa had before agreeing to the move was with José Bordalás, Valencia's May-appointed head coach. His style of football at previous club Getafe was known to be quite direct, in possession, with the ball played up to a front two to play off flick-ons, or out wide for wingers to compete against full-backs. His Getafe side produced a high volume of accurate crosses to create chances, and without the ball employed an aggressive press. Costa, after two years working under Marcelo Bielsa, should be well accustomed to Bordalás' defensive demands.
"The coach's game is attractive, it suits my style," he said.
"I talked a bit with the coach before coming, I watched some videos of the team and I saw that they play very attractive football, very fast and I think it suits my style of play. I am a fast player who likes one-on-ones. I play for the team. I like to score goals but also assist my teammates."
Costa began training yesterday at Valencia's Ciudad Deportiva de Paterna training ground, alongside fellow new signing Hugo Duro, as the rest of the squad began a three-day break.
The La Liga table shows Costa's new club sitting third after three games and they return to action on Sunday, September 12 at eighth-placed Osasuna before a potential home debut for the winger against the might of Real Madrid a week later.
"I've never been to Mestalla and I'm impatient to play in front of the fans and feel the atmosphere of the stadium," said the 27-year-old, who joined Leeds from Wolves on a loan-to-buy deal in 2019 but struggled to make an impact after a promotion to the Premier League.
"The message that I can give to the fans is that we are going to give everything for the team, that we are going to try everything to win and we are here to fight for everything."