"Sean has been up to see us for a couple of days, and I think he indicated he would be keen to join Everton, but we still have a week or so before we have to make any decisions," Everton boss David Moyes has said.
"He's injured at the moment so we will wait until the middle of next week and then look at things again.
"At the moment we are having a look at his fitness to make sure things are right."
The sale of Davis will pave the way for Fulham boss Chris Coleman to complete the signing of Sunderland midfielder Claudio Reyna, who just has personal details to settle with the Black Cats before completing his move to Loftus Road.