Felix has registered three goal contributions to help Portugal set up a quarter-final meeting with Morocco.
But Marca says Villa have emerged as shock suitors.
And the Spanish outlet claim that Unai Emery's side are suddenly in the driving seat to sign the star.
Felix became the fourth-most expensive player in history when he made the £113m move from Benfica.
But he's failed to justify his price tag, scoring just 24 league goals in three-and-a-half years.
Furthermore, recent reports claim Felix has fallen out with manager Diego Simeone.
That has opened the door for him to leave and Marca believe the club's owner Gil Marin is already negotiating a deal with Villa.
Recently-appointed Villa boss Emery will know all about Felix first-hand having come up against him five times as Villarreal manager.
Felix remained unbeaten in those five meetings, but never managed to score against Emery's team.