Bout saw the two young stars dump rivals Manchester City out of the Champions League - and both got on the scoresheet.
However Spanish giants Real Madrid and Barcelona are also monitoring Mbappe.
Kylian Mbappe proves his skills with sensational rabona
Tiemoue Bakayoko shows off his skills playing for Monaco
The 18-year-old, who is being dubbed the new Thierry Henry, scored the opener while midfielder Bakayoko grabbed the crucial third goal.
Mourinho has made attacking reinforcements a priority this summer with Wayne Rooney expected to leave Old Trafford, while a long-term successor for Michael Carrick is also on his to do list.
GETTY
Kylian Mbappe is one of Manchester United's top targets for the summer
GETTY
Tiemoue Bakayoko scored the crucial goal to send Monaco through to the quarter-finals
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Bout, who arrived at United in the summer of 2014 as Louis van Gaal's chief opposition scout, remained at the club following the Dutchman's sacking at the end of last season.
And the Daily Star reports that Mbappe and Bakayoko were the purpose of Bout's trip.