The Cottagers refused Rangers' initial bid of around £3million but Walter Smith has increased his offer for the 23-year-old after Portsmouth boss Harry Redknapp went cold on a move for Pedro Mendes.
Smith had hoped to wrap up a deal before Thursday night's Champions League qualifying round transfer deadline but Fulham are sticking to their £4million valuation.
However if Rangers can not complete the signing of Davis it is believed that they will turn their attentions to Sunderland midfielder Dickson Etuhu, who is thought to be available for around £1.5m.



