Manchester United have noticed that talks are not going well between the Premier League strugglers and the England Under-21 international, who has only 18 months left on his contract.
Sessegnon, 18, helped Fulham to win promotion last season but appears reluctant to stay if they drop back into the Championship.
Ryan, and his twin brother Steven, have been Fulham players since 2008 – joining the youth set-up as eight-year-olds.
Despite his long stay in west London, Sessegnon has seen his playing time limited under new boss Claudio Ranieri, who suggested that the left-sided player needed to be ‘more nasty’ to secure a place in his first-team plans.
The youngster made his Fulham debut as a 16-year-old in 2016 and went on to make 30 appearances in his debut season before setting the league alight the following year.
At 17, Sessegnon went on to score 15 Championship goals, which culminated in Fulham beating Aston Villa 1-0 in the Play Off final at Wembley – where Sessegnon set-up Tom Cairney’s winning goal.
His efforts didn’t go unnoticed, going on to be voted for by managers and players to win the Championship Player of the Season – all before turning 18.
Sessegnon again made history by becoming the first player born after 2000 to score a Premier League goal, which he secured against Cardiff in October.
The Roehampton-born star’s impressive performances haven’t gone unnoticed, with Fulham reportedly turning down a £25million bid from Tottenham shortly after their promotion was confirmed.
Sessegnon has played regularly in both left-back and left-midfield since his breakthrough almost three years ago, and a move to Manchester United and Tottenham was hinted at when the Fulham man revealed his footballing idols.
“When I was younger I had two players: Luke Shaw and Gareth Bale,” Sessegnon said in 2018.
“When Shaw was at Southampton he was a left-back and I loved watching him bomb up and down the wing, create goals, score goals, so I think I try to emulate that. Gareth Bale same thing really.
“At younger ages I was more attacking, and had a striker role through the middle, but as I progressed they pushed me towards left-back.
“I think they saw the energy and endurance that I have to get up and down the pitch and they just tried to change my game, more for my physical attributes than how I see the game.
“If you can play a variety of positions quite well it gives the manager the option to play you in that position, for one.
“I don’t think at a young age you know your position well enough so to have one or two is good.”
Sessegnon is likely to be just one of a number of targets in United’s sights for the upcoming summer transfer window.
But Ed Woodward and co. are also battling to keep hold of one of the club’s star turns in David de Gea.
The goalkeeper is out of contract at the end of the 2020 season, with Juventus and Paris Saint-Germain sniffing around.
Express Sport understands De Gea wants United to hand him a £350,000-a-week deal to convince him to stay put.