The goalkeeper starred for winners Portugal at Euro 2016.
Ronald Koeman has already signed Dutch goalkeeper Maarten Stekelenburg from Fulham in the transfer window, but with only Joel Robles left as a senior keeper before the start of the summer, the Dutch boss wants even more competition between the posts.
The Toffees had been linked with a move for Newcastle United stopper Tim Krul, but in recent weeks it has been suggested that they have turned their attentions towards Sporting Lisbon star Patricio.
The 28-year-old was voted the best goalkeeper at Euro 2016 for his prominent role in Portugal's first-ever major tournament victory, starring in numerous games and keeping France at bay with some fine stops in the final.
Those performances are likely to have increased interest in the man who has spent his entire career to date with Sporting, but it appears as though Everton could be at the front of the queue.
According to a report from the Sun, Patricio is keen on a move to the Premier League and could be available for as little as £6 million, which would surely represent a bargain for a keeper of his quality.
Patricio has made 373 senior appearances for Sporting during a 10-year professional career, and also boasts a vast amount of experience in the Champions League and Europa League.