The 28-year-old will be out of contract in the summer of 2022.
Spanish outlet Fichajes claims Pellegrini wants to bring the Brazilian to the Benito Villamarín.
Everton value Bernard at 12 million euros, but they could accept less given his expiring contract.
For Bernard, he wants to play in Europe next season and will be keeping an eye on Betis' situation.
The Verdiblanco are currently in sixth-place, which would see them qualify for the Europa League.