The Blues have been encouraged by talks with Barry's representatives.
Everton can ferry Barry away from Villarreal without much issue, thanks to the €40 million (£33.7m) buyout clause in the player's deal.
Patrick Boyland of The Athletic states Everton are now ready to move for the young forward, while journalist Matteo Moretto also revealed the club's interest.
VfB Stuttgart striker Nick Woltemade is also reportedly being discussed.