The deal for the 23-year-old Netherlands international is worth 39 million euros, plus an additional 5 million euros in add-ons, Ajax said.
In his breakthrough year at United last season, Greenwood made 50 appearances and scored 18 goals.
The Manchester United captain also denied bribing the police.
Dean Henderson will compete with David De Gea for the top goalkeeping role at Manchester United from 2020/21 season.
Maguire was handed a 21-month suspended sentence on Tuesday following his arrest after a brawl last week on the Greek island of Mykonos.
Maguire, who was released from custody over the weekend, is not expected to be present at the trial on the nearby island of Syros.
Paul Pogba will remain at Manchester United and discussions over a new contract will begin soon, his agent Mino Raiola said.
Maguire and the two other defendants were charged with multiple instances of assault and battery in regard to a fight they had with others on Mykonos.
Maguire, one of the most high-profile soccer players in England, was pictured being escorted into a police station in the Aegean island of Syros, which neighbors Mykonos, by plain-clothed officers on Friday.
The fixtures for the new season were released on Thursday, with Manchester City and Manchester United not playing on the opening weekend to give them more rest after their involvement in European competition this month.
Manchester United were beaten 1-2 at the semi-final stage for a third time this season, this time Sevilla ending their run with a come-from-behind victory.
Sevilla entered the final of the Europa League after defeating Manchester United 2-1.
Manchester United vs Sevilla Live Streaming Europa League in India: Here are the details of when and where to watch MAN UTD vs SEV live football match online and on television.
Dortmund sporting director Michael Zorc poured cold water on any potential deal on Monday by stating Sancho will be at Signal Iduna Park next season.
Manchester United secured a 1-0 win in extra time through a Bruno Fernandes penalty in the quarterfinal of the Europa League on Monday.
Fernandes blasted the ball past the otherwise excellent goalkeeper Karl-Johan Johnsson to give United the 95th-minute lead.
Borussia Dortmund's sporting director Michael Zorc said on Monday that Dortmund had already given Sancho a pay rise “to match the development of his performances” last summer.
Martial scored his 23rd goal of the season against LASK in last week's Europa League last-16 victory.
Manchester United vs Copenhagen Live Streaming Europa League: Here are the details of when and where to watch MAN UTD vs COP live football match online and on television.
Although named in a 30-man squad for the Europa League, Jones will be unavailable when Man United take on FC Copenhagen in the quarterfinal on Monday.
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