The former Celtic forward impressed in a 2-1 friendly win over Arsenal on Sunday, scoring both goals for his side in the second half to complete a comeback victory.
L’Equipe reports that the 23-year-old has impressed Old Trafford bosses and the Premier League giants will now make a push to secure his signing this week.
Romelu Lukaku has not played in pre-season for United and has been left out of their 26-man squad for a friendly against Norway's Kristiansund.
However, France Under-21 international Dembele has a price tag of more than £40 million and has no intention of leaving according to reports.
Dembele only joined Lyon from Scottish champions Celtic last summer and enjoyed an impressive debut season, scoring 23 goals in 52 appearances.
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