Both clubs are keen on landing the Frenchman, who is valued by Lyon at around £60million, according to the Sun.
Chelsea have made signing a striker a top priority after a difficult January window in which they had a £33m bid knocked back for Dembele.
United have held a longer interest in the player since the back end of last season and continue to monitor his situation in France.
The Red Devils are yet to replace Romelu Lukaku following his departure last summer, and are reportedly lining up a £150,000-per-week contract offer to tempt Dembele to Old Trafford.
Much could depend on how the coronavirus crisis impacts clubs economically.
United already have Odion Ighalo on loan as back-up, and are believed to be keen on making the deal permanent.
Ighalo would represent a cut-price alternative for United, which could leave the way clear for Chelsea to swoop for Dembele.
The Blues are losing Olivier Giroud when his contract expires this summer, and Michy Batshuayi could also be offloaded.
That would leave Tammy Abraham as the only recognised striker at Stamford Bridge, while Frank Lampard has the backing of club chiefs to spend in the transfer market following two windows without any purchases.
Dembele has been in decent form for Lyon so far this season, averaging almost a goal every two games over 37 appearances.
Chelsea remain interested in Red Bull Leipzig’s Timo Werner however, and could be persuaded to activate the German’s £52m release clause if a deal for Dembele proves too costly.
Lyon are believed to be prepared to sell Dembele providing their asking price is met, but are not willing to allow £65m-rated Liverpool and Man City target Houssem Aouar leave the club.