Everton boss David Moyes is prepared to make a £4million offer for Billard in the January transfer window, after failing to sign him in the summer.
Bullard joined Fulham from Wigan for £2.5million two years ago.
A Fulham insider told News of the World: “Jimmy has already had a more than decent career but who knows what more there is to come?
“For a player of his abilty it’s a shame he has never experienced European football and this should be his next step.
“Everton can offer that if they survive in the UEFA Cup until the New Year and the prospect of that might overcome any reservations he may have of returning north from his London base.”
