The Ibrox club have agreed a fee of £3million for the Northern Ireland international.
Scottish Premier League win over Hearts that Davis is now close to putting signing.
He said: "We have reached agreement with Fulham and we still have one or two things to settle with the player and his representatives but we are hopeful that one will go through.
"That should happen in the next couple of days.
"It's a transfer that has been really ongoing so we don't want to count our chickens.
"But we have agreed with the club now, which is a big thing, and we would hope we can reach agreement with Steven's people over the next couple of days."