New Chelsea manager Xabi Alonso is pushing for the club to sign the 28-year-old as their replacement for Marc Cucurella.
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Chavarria played a key role for Rayo Vallecano last season, making 44 appearances across all competitions as they finished in eighth and reached the Conference League final.
According to Marca, Cheslea have agreed terms with the left-back, who is keen on the move, but not yet with his club.
It’s understood that Chelsea’s latest bid was worth £10, but the Spanish club are holding out for a fee of £15 million.