The 27-year-old has informed Chelsea of his desire to leave Stamford Bridge this summer, according to sources, amid interest from both Spanish clubs.
Cucurella was critical of the direction of Chelsea's squad in a March interview during the international break, telling The Athletic following the club's Champions League exit to Paris Saint-Germain: "I understand this is part of the club's policy, and that they want to take this direction — signing young players and looking to the future.
"But, for all of us who are still here and want to win big things, moments like this make you feel discouraged."
Speaking to El Partidazo de COPE ahead of Spain's opening Group fixture against Cape Verde, Cucurella was asked directly about his future and offered a deliberately vague response.
"Right now, I don't want to talk about that. I'm very happy at Chelsea, but for some time now I haven't wanted to know anything about those discussions. I want to enjoy this experience because it could be my first and my last World Cup. I just want to enjoy it."
When pressed specifically on whether he would prefer a move to Barcelona or Atletico Madrid, Cucurella declined to engage further.
"No, no. I'll pass."
Cucurella joined Chelsea from Brighton & Hove Albion in 2022 for an initial fee of £56 million and has become one of the more reliable performers in a turbulent period for the club, making 50 appearances and contributing five goals last season despite Chelsea's underwhelming campaign. He has three years remaining on his contract.
Atlético Madrid's interest stems from a perceived gap at left-back, with Matteo Ruggeri currently their only out-and-out option in the position and himself linked with a move away from the Estadio Metropolitano. Barcelona's interest is reportedly contingent on the future of Alejandro Balde, with any approach for Cucurella dependent on Balde's situation resolving first. Manchester United have also been credited with interest as they search for a long-term successor to Luke Shaw.
Chelsea sources indicate the club is not actively pushing to sell Cucurella, but would not block a departure if an offer meets their valuation, reported to be in excess of £40 million.
For a player who has previously joked he would rather shave his trademark curls than join Real Madrid — while leaving the door open to an eventual Barcelona return — Cucurella's evasiveness on Friday suggests the World Cup, not the transfer market, is exactly where his focus currently sits. Spain begin their campaign as one of the tournament favourites, and Cucurella starts at left-back regardless of where he plays next season.