A deal worth €17.5 million was agreed between the two clubs on Saturday, with personal terms and a medical having now been finalised.
The Netherlands international, who can play both in midfield and at left-back, bid farewell to his Ajax team-mates, with whom he won four consesutive league titles.
The son of Ajax and Netherlands legend Danny Blind had earlier expressed his delight at having secured a switch to Old Trafford, telling reporters that United are "one of the biggest clubs in the world".
His acquisition by Louis van Gaal's side, along with British-record signing Angel Di Maria, has been welcomed by new team-mate Juan Mata, saying: "I am sure these two players will give a lot to us and will strengthen the team.
"We already know what Angel is capable of, his ability and dribbling skills, and Daley has a great touch and is very intelligent, as he showed during the World Cup."
However, Red Devils legend Paul Scholes is less convinced that Blind will improve the side.
Blind was a key member of the Netherlands team who, under Van Gaal, finished third at the World Cup over the summer.