Mundo Deportivo report that Rayo Vallecano turned down the initial offer, but a transfer has not been ruled out, with Chelsea expected to return with improved terms and a deal described as likely.
Marca first reported Chelsea's interest last week, with incoming manager Xabi Alonso understood to have identified Chavarría personally as his top target for the position — a left-back who has been a fixture in La Liga with a club the Spaniard knows well from his time in the country.
Chavarría, 28, was born in Figueres, Catalonia, and worked his way up through Spain's lower leagues before earning a move to the professional game.
He spent two years at Real Zaragoza in the Segunda División, making 74 league appearances and establishing himself as one of the division's more reliable full-backs, before Rayo Vallecano signed him on a five-year deal in August 2022.
He has been a mainstay at the Estadio de Vallecas since, making 125 appearances in total across his four seasons at the club.
Last season he produced one goal and two assists in 33 La Liga starts — his most settled and consistent campaign in the top flight to date.
His contract runs until 2030, handing Rayo significant leverage in any negotiation, and the Spanish club are in no hurry to sell cheaply.
Bayer Leverkusen are also said to have registered interest in the defender, and Chelsea are aware that moving slowly risks losing him to a rival.
Chelsea's need for a specialist left-back is genuine. Cucurella made 163 appearances for the club and was central to their recent successes, including winning the Conference League and Club World Cup, and finding a like-for-like replacement at short notice presents a real challenge.