United will pay a loan fee of €10 million ($10.8m), with an option to make the move permanent next summer for €20m plus €5m in add-ons, sources told ESPN.
Sources have told ESPN that United were only able to make a move for Amrabat, 27, after agreeing Dean Henderson's move to Crystal Palace. Henderson joined Palace on Thursday in a deal which could rise to £20 million ($25.4m).
"It is a huge honour to become a Manchester United player," Amrabat said. "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."
Added United football director John Murtough: "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."
Meanwhile, left-back Sergio Reguilon is set to be available to face Arsenal on Sunday. He passed a medical on Thursday night and trained with United on Friday morning.
Rasmus Højlund could make his debut against Arsenal after coming through his first week of training with the first team. The £72m striker is set to be part of the squad at the Emirates and speaking at his news conference on Friday, manager Erik ten Hag said the 20-year-old is fit enough to start if needed.
"He had a good training week and there is final training tomorrow [Saturday]," Ten Hag said.
"He's doing well and he will be available. I think for every player there is a big spotlight. At United you have to perform, every position, every player
"I think we have done good business, we constructed a strong squad and we are ready for the fight."