The Premier League sides are reported to be on the hunt for a striker this month as they look to equip themselves for a tough second half of the season.
Chelsea are keen to give Tammy Abraham some competition, while they also look to offload Olivier Giroud.
Manchester United's needs are more immediate, with the club recently losing Marcus Rashford to a back problem that could see him sidelined for three months.
The Red Devils are reluctant to rely on Anthony Martial, who missed a golden opportunity in the 2-0 loss to league leaders Liverpool on Sunday evening.
Mason Greenwood is seen as a star of the future, but Ole Gunnar Solskjaer doesn't want to put unnecessary pressure on the 18-year-old.
According to Foot Mercato (via Sportwitness), United are ready to greenlight a bid for Dembele in the coming days.
The bid is understood to be in the region of €50m plus performance-based bonuses.
Solskjaer alluded to the squad requiring a striker before the January window opened, but so far no deal has materialised.
United reportedly watched Dembele in action for Lyon against Nantes on the weekend and were sufficiently convinced now is the time to make their move.
Lyon president Michel Aulas has said he won't sanction the sale of Dembele in the winter window, but his words are always taken with a pinch of salt.
The 23-year-old has scored 11 goals in 19 games in Ligue 1 so far this season.
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