After months of negotiations, Fulham think they may be able to wrap up Greening's signing this week.
Greening has been the subject of dispute between West Brom and Fulham, with The Baggies rejecting all offers thus far, including the most recent £2million bid.
However, according to The Daily Mail, The Cottagers are confident that a £3.25million bid will be too good to refuse, and are expected to table the bid in the near future.
Greening has stated his intention to play in the Premier League throughout the summer, although he is willing to remain at The Hawthorns if West Brom do not sell him.