Cucurella accepted a €50M switch to the Spanish capital ahead of playing a vital role in La Roja's title win at the 2026 FIFA World Cup.
He is expected to link up with his new teammates in mid-August, following a belated summer break, as Chelsea begin their search for a replacement at Stamford Bridge.
Reports from Mundo Deportivo claim the two clubs come into direct contact again with Real Madrid prepared to 'do Xabi Alonso a favour' amid interest from their former boss in Alvaro Carreras.
Cucurella's arrival is likely to keep Carreras as second choice at left-back - despite Fran Garcia's move to Real Betis - with Ferland Mendy still sidelined through injury.
Carreras was highly rated by Alonso, following his arrival from Benfica last summer, and he will start the season in place as Cucurella builds up his match fitness.
However, Real Madrid could be open to a €45M deal, which would effectively cover the majority of their original outlay on Cucurella and give both teams depth on the left side of their defences.