Hull City are at the top of the Championship table at the moment.
West Ham, Swansea, Hull City and Sheffield Wednesday are also keen on the Scotland international, according to reports.
The Hammers have already beaten Jurgen Klopp's side twice in the Premier League.
Everton manager Roberto Martinez addressed the media ahead of his side's FA Cup clash with Carlisle United this weekend, on the back of a defeat in the League Cup semi-final.
Chris Dawson wasted little time finding a new club after leaving Leeds United.
Former Gers striker Kris Boyd questioned whether Warburton would stick around if King failed to bring their targets to the club.
The Italian side are keen on Federico Fazio and Angelo Ogbonna.
With fixtures continuing to come thick and fast, Jurgen Klopp may make changes again.
Wolverhampton Wanderers have snapped up striker Joe Mason from Cardiff City.
The Derby defender played for the Premier League's bottom side for over three years.