Chavarria is a more attacking option than Cucurella, but sources suggest he could fit perfectly into Alonso's preferred three-centre-back system.
Several reports indicate that personal terms between the London club and the Catalan defender are effectively agreed, with the real sticking point being negotiations with Rayo Vallecano over a fee.
Chelsea are understood to have tabled an offer of around €15million, well short of the €50million release clause inserted into the 28-year-old's contract.
According to Fabrizio Romano, Chelsea remain determined to get the deal done and continue working on it despite the rejection of their opening proposal.
Talks are said to be ongoing, with a fresh meeting expected in the coming days that could prove decisive in Chavarria's move from Madrid to Stamford Bridge.
Chavarria did not make his La Liga debut until he was 24, and his European bow came at 27, but he has since become one of the division's most reliable full-backs, making 125 appearances for Rayo after joining from Real Zaragoza for €1.8million in 2022.
Negotiating with Rayo Vallecano president Raul Martin Presa is rarely straightforward, and Chelsea know it.
The Spanish club did something similar last year with Andrei Ratiu when English football came calling, insisting on the release clause before letting the Romanian leave.
Now, the situation looks set to repeat itself with Chavarria, and Rayo are even said to be considering reporting Chelsea to FIFA for allegedly negotiating with a player still under contract without going through the proper channels.
Cucurella's own move to Real Madrid was completed for £51.8million last month, leaving Chelsea with the funds and the motivation to move quickly for his eventual replacement.
Alonso, who officially takes charge at Stamford Bridge this month, appears to have already identified his man; the question now is simply whether Chelsea and Rayo can find common ground on a fee.