The Toffees are believed to be in the market for additions ahead of the transfer deadline, with manager David Moyes keen to build on an impressive start to the Premier League campaign.
Venezuelan international Amorebieta is thought to be a man the club have looked at ahead of a potential swoop, and it appears that he would be willing to make a move.
According to reports, the 27-year-old has become frustrated at Bilbao after watching a number of top stars leave the club.
Javi Martinez Joined Bayern Munich last summer, whilst Fernando Llorente looks set to depart to sign with Juventus at the end of the season.
This has caused Amorebieta to consider his own future, with the centre-back believed to feel that Athletic are moving in the wrong direction.
With their Basque only policy, the Spaniards will not be in any hurry to sell, but a significant bid may alter their stance.
Athletic have reportedly offered Amorebieta a new five-year contract worth around £8.5m, but he is thought to be stalling in an attempt to up interest before the transfer window closes.
Despite representing Venezuela, he holds Spanish citizenship meaning that Everton would not need to apply for a work permit.