King is out of contract in the summer and keen to make sure he is playing regularly in the Premier League for the second half of the season.
Both Everton and Fulham have had offers accepted, but it is claimed that the Cottagers may come out on top due to wanting King on a permanent deal.
Fulham want to sign King outright, while Everton have only offered to take King on loan until the end of the season.
If King snubs Everton then Carlo Ancelotti will find himself with reduced attacking options.
Everton loaned Moise Kean out to Paris Saint-Germain in the last transfer window, while this month they have allowed Cenk Tosun to link up with Besiktas.
Winger Anthony Gordon has joined Preston North End, further reducing the attackers Ancelotti has to call upon.
It remains to be seen if Everton can work their magic and convince King to move to Merseyside.
