The Sun is reporting that the 17-year-old defender could be willing to stay with the Cottagers if they manage to secure promotion from the Championship this season.
The same story says that Sessegnon has attracted admiring glances from Manchester United, Manchester City and Tottenham ahead of the summer transfer window.
However, the report says that the teenager has “made it clear” that he would be willing to stay at Craven Cottage beyond the summer if they can provide him with top-flight football next season.
Fulham are currently in third place in the Championship as they seek promotion back to the Premier League for next season.
The teenager has been in superb form for the Cottagers so far this season, scoring 14 goals and making five assists in 38 Championship games for the London side.
Manchester United, who are second in the top flight, are likely to be on the lookout for a number of new signings this summer as Jose Mourinho looks to assemble a squad capable of winning the Premier League next season.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news