The deal is reportedly worth an initial £34.6m with £4.4m worth of add-ons.
The Netherlands international has signed a five-year contract with his new club.
Speaking after the announcement, he said: "I cannot begin to explain how incredible an opportunity it is to join a club with such an amazing history. I would like to thank everyone at Ajax, I grew up there and I will always have a special bond with the club. I am now ready to take the next step in my career and perform at the highest level and there is no higher standard than Manchester United.
"Everyone has told me how amazing the Old Trafford atmosphere is and I am looking forward to experiencing that, once it is safe for the fans to return.
"This team has some of the best midfielders in the world and I know that I can learn from them and also bring my own strengths to the group. Having spoken to the manager about his vision for this team, the direction that the club is going in is hugely exciting and I cannot wait to be a part of that."
Hope you're well.
Just announcing @Donny_Beek6.
Regards, @ManUtd. pic.twitter.com/Exq6b2M5jy
— Manchester United (@ManUtd) September 2, 2020