Having arrived from Paris Saint-Germain in 2012, the 19-year-old forward is beginning to impress on a more regular basis in west London.
Dembele netted his third goal of the campaign as Fulham drew 1-1 with Leeds on Wednesday, producing a display that will only lead the interest in his services to grow.
Everton are reported to be amongst those tracking the teenager, who Press Association Sport understands is close to committing his future to the Craven Cottage club.
Dembele’s current deal expires at the end of the season but in the next few weeks looks set to a sign a four-year extension, keeping him in west London until 2020.
The news would be a welcome boost for the Cottagers after losing another talented prospect in the summer, when Patrick Roberts made a big-money move to Manchester City.