The Old Trafford club sent scout David Friio to watch the 20-year-old play and score twice for France’s Under 21 side against England on Monday.
Manager Jose Mourinho is understood to be a fan of Dembele and watched him three times last season, when he played for Fulham before his move to Parkhead.
The Portuguese is thought to see similarities between the youngster and former Chelsea charge Didier Drogba. Sportsmailunderstands Celtic do not want Dembele to leave in the January window.
Dembele’s value has rocketed from the £500 000 he cost Celtic in June, but he is not keen to leave Parkhead after just six months, during which time he has blossomed under the guidance of manager Brendan Rodgers.
United may choose to take a long-term approach, but their desire to move in January would be increased if winger Memphis Depay were to leave Old Trafford, possibly to Everton on loan.
Manchester City, Arsenal, Chelsea and Bayern Munich also sent scouts on Monday to watch Dembele.
United have been aware of the player’s talent since his time at Paris Saint-Germain and their interest was firmed up by a performance against United’s Under 21s in October 2013. That game marked Darren Fletcher’s return to action after 10 months out and was watched by Nicky Butt, who is currently the head of coaching at Manchester United’s academy and then first-team coach Phil Neville, with club icon David Beckham also present. Dembele scored for Fulham in the 36th minute of the match.
He has made a blistering start to life at Parkhead, scoring three times in the Champions League group stage - including two in the 3-3 draw with Manchester City - and seven times in the Scottish Premiership.
He also scored the first Old Firm league hat-trick since 1966 as Celtic beat Rangers 5-1 on September 10.
Original source: Mourinho eyes a move for goal ace Dembele