No financial details were disclosed for Amrabat's move but British media said United will pay 8.6 million pounds ($10.83 million) and there is an option to buy.
"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," he said.
Amrabat, who has 49 caps, played every minute of Morocco's run to the World Cup semi-finals in Qatar last year.
Reguilon will arrive as a replacement for Luke Shaw, who will be sidelined for several weeks after sustaining a muscle injury last month. Back-up left back Tyrell Malacia is also sidelined due to an injury he picked up last season.
"Having spoken to the manager, I know what he needs from me, and I am ready to play my part in helping the team to achieve success," Reguilon said.
"I know that I can contribute to Manchester United this season, I am ready to fight for this group and show everyone my qualities."
He made over 60 appearances for Spurs in his first two seasons in England, but spent the 2022-23 campaign on loan at Atletico Madrid.
United, with two wins from three league games, travel to face Arsenal on Sunday, with the Gunners sitting one point ahead of them in the table.