The striker was a target of both Chris Hughton and Alan Pardew, but new Fulham boss Martin Jol is looking to land him.
The Republic of Ireland striker appears to have little future at White Hart Lane and was expected to complete a move to West Ham for around £6m had they not been relegated.
Once they went down that ended that move but former Spurs boss Jol is now ready to be reunited with Keane as he looks to add to Fulham's front line.
£4m could be enough, but there may still be a stumbling block over his wages.
Keane is thought to be earning around £70,000 a week at Spurs and that may prove too rich for Fulham.