The Red Devils' £89million swoop for Paul Pogba from Juventus last summer currently stands as the biggest transfer fee.
But Jose Mourinho appears ready to top that as he goes in search of a long-term solution up front at Old Trafford.
Despite Mbappe being just 18 years of age, the French forward has built up a strong reputation already this season.
Kylian Mbappe's top form at Monaco has earned him a call-up to France's senior squad
The Red Devils have tabled a staggering £95m
His form for Monaco in Ligue 1 and the Champions league has attracted interest from United, Arsenal and Real Madrid.
And according to Spanish radio station Cadena Cope, the Red Devils have tabled a staggering £95m.
Atletico Madrid hitman Antoine Griezmann is thought to be Mourinho's No1 target this summer, but attentions seem to have turned to Mbappe.
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Prior to United's latest offer, a £69m bid was also put forward and turned down by the French outfit.
The £95m approach is said to have come after Manchester City's Champions League round of 16 first leg clash with Monaco where Mbappe scored at the Etihad.
He then netted in the second leg to ensure Monaco progressed on away goals to the quarter-finals, highlighting his huge potential.
Leonardo Jardim's side are reluctant to sell though, shown by their rejection of a world-record bid for Mbappe.
Cadena Cope add Madrid are yet to make an offer but remain keen on luring the teenager to the Bernabeu.
At the moment, Manchester United are apparently the only team to have made a firm move.