The London club have slapped a £40million price tag on the teenager, who will turn 19 on Saturday.
Tottenham and Juventus have also been linked with Sessegnon, who can play as a left back or winger.
Manchester United have made initial contact with Fulham over the potential signing of Sessegnon as Ole Gunnar Solskjaer wants to recruit young, promising players under the age of 24 this summer.
Tottenham boss Mauricio Pochettino has gone on record to say he wants new signings to arrive as early possible in order to integrate them into his squad as quickly as possible ahead of the season.
Spurs chairman Daniel Levy, however, has a reputation as someone who prefers to strike deals towards the latter stages of transfer windows.
But the north London club have changed tack this summer with a deal for Sessegnon already in the offing, while they have also made overtures regarding Barcelona’s Andre Gomes, who spent last season on loan at Everton.
Spurs are in pole position to land Fulham youngster Sessegnon, though Paris Saint-Germain retain a serious interest.
Craven Cottage chiefs are aware of Tottenham's interest in the talented left-footer and now Fulham’s season is over, Spurs will look to conclude a swift deal.
Fulham value Ryan at £40million - but Tottenham want to pay closer to £25million.
The 18-year-old has been a primary target for Spurs for almost two years, but the Champions League finalists are growing increasingly confident of finally getting a deal over the line.
Sessegnon wants to remain in the Premier League, with a move to Tottenham understood to be his preferred destination.