The PA news agency understands United have paid a 10million euros (£8.6m) loan fee for Amrabat, with the season-long deal including the option to trigger a permanent move.
If that option is taken, Fiorentina will receive a 20m euros (£17.1m) transfer fee and up to 5m euros (£4.3m) in potential add-ons.
Morocco midfielder Amrabat said: “It is a huge honour to become a Manchester United player. I’ve had to be patient for this moment but I’m someone who always listens to my heart and now I am representing the club of my dreams.
“I am a passionate player; I want to bring that energy to the squad, and I will put everything into every action I take for the team.
“I know exactly how Erik ten Hag likes to work and what he needs from his players.
“His coaching and guidance developed me so much as a player earlier in my career; I know that he will help to bring out the best of my abilities so that I can help the group to be successful this season.”
Amrabat played under Ten Hag at Utrecht earlier in his career before moving on to Feyenoord, Club Brugge and then Fiorentina.
The 27-year-old has won 49 caps for Morocco and played every minute of his country’s run to the semi-final of the 2022 FIFA World Cup.
Amrabat’s arrival was viewed as essential by United fans given their early-season problems in that area with opponents being able to attack their back four too easily.
United football director John Murtough said: “We have tracked Sofyan for a long time, so we are really pleased to bring him to Manchester United.
“His committed, high-energy approach to the game fits perfectly with the group that we are building here.
“We know that Sofyan’s mentality, dynamism and technical qualities will help the squad as we look to achieve success in all competitions this season.”
Earlier on Friday, United confirmed the arrival of Turkey international goalkeeper Altay Bayindir from Fenerbahce and defender Sergio Reguilon on loan from Tottenham until June 2024.
Bayindir, 25, has signed an initial four-year contract at Old Trafford, with the option of a further year, and will provide competition to fellow new arrival Andre Onana.