The deal for 22-year-old Brazil international midfielder Santos is worth 50m pounds, with United set to pay 48m upfront plus 2m in add-ons.
Santos said he was looking forward to working with United manager Michael Carrick, who himself was a top midfielder in his playing days.
In quotes published on United's website, Santos said: "Everything about Manchester United is special; it is an incredible feeling to join a club that some of my biggest idols have represented.
"As a midfielder, I am really excited to have the opportunity to learn from Michael Carrick; he is the perfect coach to help me take the next step in my career and push to achieve my dreams.
"Everybody told me about how ambitious the club is and the amazing environment that has been created here. I know just how strong the squad is and I cannot wait to fight together to compete for the biggest trophies."
Jason Wilcox, United's director of football, added: "Andrey is an outstanding midfielder with excellent technical qualities and the ability to impact the game at both ends of the pitch.
"He was a key target for us in an important position, so we are delighted that he will join Michael's squad so early in pre-season."
United will continue to stack up their midfield options with reports suggesting that the Premier League giants have activated the release clause of Belgian international Tielemans.
Belgium international Youri Tielemans said he "feels incredible" after penning a five-year contract to join Manchester United from fellow Premier League side Aston Villa.
Manchester United have strengthened their goalkeeping options by bringing in experienced Wales international Karl Darlow on a two-year deal, with the agreement including the option of an additional 12 months.
Brighton have completed the signing of Croatia defender Luka Vuskovic from Tottenham Hotspur in a club-record deal worth up to 50 million pounds.
Andoni Iraola believes his coaching journey has prepared him for the challenge of leading one of world football's biggest institutions after taking charge of Liverpool on a two-year deal.
