The Red Devils had chief scout Marcel Bout in the stands when Mbappe and Bakayoko scored for the French Ligue 1 side against Manchester City in a Champions League round of 16 second leg tie on Wednesday, March 15, according to ESPNFC sources.
Spanish giants Real Madrid and Barcelona are also reportedly keeping an eye on the 18-year-old Mbappe.
Manchester United manager Jose Mourinho wants to sign attacking players this summer with Wayne Rooney expected to leave Old Trafford.
United also need a long-term successor for Michael Carrick and Bakayoko, 22, can fill the void.
Monaco vice-president Vadim Vasilyev though said on Wednesday that they would like to hold on to Mbappe.
Meanwhile, Mbappe said after the match that Monaco's dominant first-half show was the bedrock to their success against City.
"We really wanted to show them from the start that it wasn't going to be their day and that it would well and truly be ours.
"We managed to establish ourselves in their half of the pitch, we didn't let them play. Almost every chance was turned into a goal," he said.
"I think it was a real blow to their morale when they went in 2-0 down at half-time. I don't think they expected that."
Portugal international Bernardo Silva said: "At 2-1, we didn't falter. That's what has been our strength this season, we're not afraid to play."
"Even at 3-2 in Manchester, we continued playing and that's also why we conceded five goals. But this evening, we're very happy. Thanks to Bakayoko for taking us into the quarter-final."
"It's a magical evening. If there is one very positive thing this season, it's that we have a group. When a player is missing, the replacements come into the team without us losing any quality.
"Now we'll see who we'll play against. There are only fantastic teams left. We dream of going beyond the quarter-finals," he added.