British media reported that the United academy product will cost the Ligue 1 club up to 31.6 million euros.
Greenwood made 129 appearances and scored 35 goals for United after making his debut in 2019.
Greenwood spent last season on loan at Getafe in Spain after United said they wanted to avoid him becoming a distraction despite all criminal charges against him being dropped following allegations, which he denied, of assault and attempted rape.
The winger scored 10 goals in 36 appearances in all competitions for Getafe, who finished 12th in LaLiga.
Getafe President Angel Torres expressed in May his desire to keep Greenwood for another season, while LaLiga president Javier Tebas said he hoped the forward would extend his stay in Spain.
Marseille, who last month appointed Italian former Brighton & Hove Albion manager Roberto De Zerbi as head coach, begin their Ligue 1 campaign at Brest on August 18th.