Premier League champions City have splashed $65 million on Savinho - also known as Savio - who has moved from Troyes and signed a five-year contract.
Meanwhile, United and Marseille are reported to have agreed a $51 million fee for Greenwood, who spent last season on loan at Spanish side Getafe, and who has also inked a five-year deal with his new club.
Savinho, 20, spent last season on loan at another of City's sister clubs, Girona, and helped the team qualify for next season's Champions League.
"I'm so happy to be joining Manchester City, the Premier League champions and Club World Cup winners. Everyone knows they are the best team in the world right now, so to be here is very exciting for me," the player said in a statement.
Savinho was one of the stand-out players in the Spanish league last year, scoring 11 goals and providing 10 assists.
He was part of Brazil's squad for the Copa America and has played seven times for his country.
Mason Greenwood has left Manchester United to link up with French League side Marseille. (EPA PHOTO)
Greenwood was suspended by United on January 30, 2022 over allegations relating to a young woman after images and videos were posted online.
The one-cap England international faced charges including attempted rape and assault, but the Crown Prosecution Service announced in February last year that the case had been discontinued.
A statement from the French side read: "Olympique de Marseille are delighted to announce the signing of Mason Greenwood from Manchester United. The 22-year-old striker has signed for the club following a successful medical."
United confirmed the move in a statement, adding: "We wish Mason all the best in his future career."
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