It is also claimed Manchester City have tabled an offer worth £104m for the Monaco star.
Mbappe is the hottest property in Europe this summer after a stunning breakthrough season in Ligue 1.
Mbappe, 18, shone as Monaco won the title and reached the semi-finals of the Champions League.
He scored 15 league goals and six in Europe and is now valued at over £100m.
Real Madrid and Arsenal have been showing interest in the youngster but Spanish newspaper AS claim United have made a stunning £113m offer.
Antoine Griezmann has decided to stay at Atletico Madrid for another season and Mourinho is now targeting a number nine style striker.
Romelu Lukaku, Alvaro Morata and Andrea Belotti are on his radar and now United have firmed up their interest in Mbappe.
But they face competition from rivals City – although they have offered more money for him.
Arsene Wenger, who is also keen on the striker, has advised the Monaco man to pick his next club wisely.
“Somewhere where he has a good chance to play and is sure to play.
“That is why I think Deschamps made that statement.
“He thinks that he needs to play but I would say as well on the other hand that the boy has made enough impression to be basically sure a club who spends a 100 million for you is basically to make at the start at least the red carpet.
“Nobody will buy a player for a 100 million and say, “Come on, sit in the stands.”