Six weeks after their first offer was rebuffed, Ruben Amorim's side have agreed terms with fellow EPL side the Bees to sign the 25-year-old forward.
United are set to pay an initial £65m (A$134m) for Mbeumo, plus potential add-ons.
The Cameroon international will become the Premier League club's second major summer signing after Brazil international Matheus Cunha, who joined from Wolves last month.
Mbeumo expressed a firm desire to move to Old Trafford over other clubs, including Tottenham, and his arrival is a major boost ahead of United's pre-season tour.
Amorim's men fly to the United States on Tuesday for the three-game Premier League Summer Series.
Mbeumo has just had the best season of his career, scoring 20 Premier League goals and providing seven assists.
Meanwhile, Arsenal have completed the £48m (A$99m) transfer of Chelsea forward Noni Madueke, who becomes Mikel Arteta's fourth signing of the summer.
The 23-year-old is understood to have signed a five-year deal. The England international will provide competition to Bukayo Saka and Gabriel Martinelli in wide areas.
However, Madueke's move across London has been met with severe criticism by Arsenal supporters on social media.
Madueke left Chelsea's eventual Club World Cup winning campaign in the United States last Friday to undergo his medical at Arsenal and put the finishing touches on his controversial move.
Madueke scored 11 times for Chelsea last season and was rewarded with his first senior international appearances under England interim manager Lee Carsley.
Arsenal remain hopeful of signing striker Viktor Gyokeres from Sporting Lisbon, although it is not believed that a deal is imminent.
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