The 36-year-old Italian has been without a club since parting ways with Botafogo in December, a role that represented his first independent head coaching position after years as an assistant to his father Carlo Ancelotti at Napoli, Everton and Real Madrid.
Radio journalist Raúl Granado of Onda Cero reported that the club president's first choice is Ancelotti ahead of other names on the shortlist including Jagoba Arrasate and Imanol Alguacil, both of whom have wider support within the club hierarchy.
Ancelotti is believed to be keen on a return to Madrid, a city where he lived for several years during his time as assistant at the Bernabéu. The club reportedly view him as a coach in the mould of Pérez's tenure — young, progressive and still developing his own coaching identity.
No formal offer has been made. Pérez's future will need to be resolved first, with Villarreal yet to make an official approach.