The Blues were linked with a move for the Portugal international last season and could make a move for him this summer.
But Manchester United have also registered an interest while Ligue 1 side Monaco have already made an approach for the winger.
The 22-year-old has impressed many since his move to the Archbishops in 2013 and has seven goals and seven assists for the Liga NOS outfit this season.
Silva is believed to have a £15.5million release clause in his current contract with SC Braga and has been capped five times for his national side.
Everton have a number of widemen on their books, Gerard Deulofeu, Aaron Lennon, Kevin Mirallas, Oumar Niasse, Steven Pienaar, Tom Cleverley and Aiden Mcgeady, (currently on loan at Sheffield Wednesday) are all capable of playing out wide, but Silva is a left winger.
The Blues have struggled to find a long-term replacement for Pienaar on the left hand side, but Silva may be the man to fill the diminutive South Africans position for good.