The fee eclipses the 81 million euros Liverpool paid Southampton for Virgil van Dijk in 2018.
"I am delighted to have signed for this great club," Maguire said on United's website.
"I have thoroughly enjoyed my time at Leicester and would like to thank everyone at the club. However, when Manchester United come knocking on your door, it is an incredible opportunity."
Maguire is the third arrival of the summer at Old Trafford. United boss Ole Gunnar Solskjaer has drafted in Daniel James from Swansea City and Aaron wan-Bissaka from Crystal Palace.
"From my conversations with the manager, I am excited about the vision and plans he has for the team," Maguire added.
"It's clear to see that Ole is building a team to win trophies. I am now looking forward to meeting my new team-mates and getting the season started."
Maguire could make his debut in his new club's Premier League opener at home to Chelsea on 11 August.
"Harry is one of the best centre-backs in the game today and I am delighted we have secured his signature," Solskjaer said.
"He is a great reader of the game and has a strong presence on the pitch, with the ability to remain calm under pressure - coupled with his composure on the ball and a huge presence in both boxes.
"I can see he will fit well into this group both on and off the pitch. He has a great personality and is a fantastic addition to the club."
Premier League champions Manchester City had been interested in Maguire. However manager Pepe Guardiola admitted on Sunday that the champions could not afford his price tag.