Mikel Arteta has been linked to the Everton job of late.
Steven Gerrard recently signed a four-year deal to become Rangers manager.
Allardyce has been the centre of criticism for Everton fans since his arrival.
Everton have been linked with a move for PSV technical director Marcel Brands.
Graeme Murty’s Glasgow Rangers lost on Sunday, and former Ibrox stars Michael Ball and Lee McCulloch were not impressed.
Everton drew with Liverpool at Goodison Park in the Premier League at the weekend.
Michael Ball has strongly criticised Everton manager Sam Allardyce.
Everton suffered a 3-1 loss to Manchester City on Saturday evening.
Cenk Tosun scored twice for Everton in the Premier League at the weekend.
Leighton Baines played for Everton against Brighton and Hove Albion at Goodison Park in the Premier League at the weekend.