The 31-year-old goalkeeper is the number 2 at Barcelona behind Marc-Andre Ter Stegen and made just 12 appearances across all competitions last season for the club.
Barcelona are not actively looking to offload Neto, but could consider letting him go for the right price.
Neto was considering leaving Barcelona in the winter transfer window after feeling frustrated at warming the bench at the Nou Camp.
And according to Spanish daily Mundo Deportivo, Everton are interested in taking Neto to Goodison Park in the summer transfer window.
They did not sign Robin Olsen on a permanent deal at the end of last season and are in the market for a goalkeeper.
The Barcelona man has emerged as an option for the Toffees, who are considering signing him.
Jordan Pickford was the first choice goalkeeper under Carlo Ancelotti, but if he joins Everton this summer Neto could have an opportunity to prove himself under a new manager at Goodison Park.
Everton are also on the cusp of bringing in a new manager with Nuno Espirito Santo now the clear favourite to succeed Ancelotti.
