The striker is wanted by both Premier League clubs this month as they look to bolster their attacking options going into the final few months of the season.
Chelsea had a bid rejected for him by Lyon last week but Sportsmail understands the Blues are still keen on the 23-year-old despite his current side saying they have no intention of selling him.
Scouts from Manchester United and Chelsea watched Moussa Dembele (R) in action for Lyon
Dembele scored a fine effort in Lyon’s 3-1 League Cup win over Stade Brest on Wednesday
He has hit 10 goals in Ligue 1 already this season and took his tally for the campaign across all competitions to 13 with a fine effort in Lyon’s 3-1 win over Stade Brest on Wednesday.
Both Chelsea and United had scouts in attendance for the game as the Blues weigh up making another bid for Dembele, while Ole Gunnar Solskjaer considers making a move as well.
Lampard is free to spend after the Blues were banned from making signings in the summer and has identified Dembele as a player who would improve his forward options.
Solskjaer, meanwhile, desperately needs another forward option having failed to sign a replacement for Romelu Lukaku in the summer.
United have been far too reliant on Marcus Rashford and Anthony Martial so far this campaign and need another regular source of goals in their pursuit of Champions League football.
Both Ole Gunnar Solskjaer and Frank Lampard are interested in signing Dembele this month
Dembele’s history — he started his professional career at Fulham before moving to Celtic and then Lyon — means he should slot effortlessly into both United and Chelsea’s squads.
Both sides were dealt a blow in their pursuits of Dembele on Wednesday though when Lyon owner and president Jean-Michel Aulas insists he will not be sold this month.
Speaking about the 23-year-old he said: ‘Moussa will not leave.
‘On one hand, he wants to stay with Lyon. It’s the best way for him to continue to improve his game.
‘There have been offers for him and other players, but no one we want to keep will leave in January.’