The Toffees have been linked with the Argentine for some time, and it is understood that scouts from the club watched him in action last week.
Fazio was a part of the Spanish side’s Europa League fixture with NK Maribor, and performed impressively.
As a result, Everton may look to push through a deal when the transfer window re-opens.
Roberto Martinez is thought to be gearing up for a big summer of activity, with a number of his current crop of players only on loan at the club.
The Spaniard is keen to bring in top class names, and is hopeful that Fazio can help his side to break into the top four.
A centre-back by trade, the South American can also operate in a wide defensive role or in midfield.
Liverpool and Arsenal have also been linked with the player, but it remains unclear as to whether or not their interest is serious.