The agreement also involved Costa Rica international keeper Keylor Navas moving in the opposite direction.
De Gea's former number two Anders Lindegaard was also heading out of Old Trafford as he joined West Bromwich Albion on a two-year deal.
There were two other high-profile departures from United. Striker Javier Hernandez, who spent last season at Real Madrid, joined Bayer Leverkusen for $16 million, and winger Adnan Januzaj moved to Borussia Dortmund on a season-long loan.
United was also active in terms of recruitment too, with negotiations over a possible $78 million deal for highly-rated France Under-21 striker Anthony Martial, 19, well advanced.
If confirmed, such a fee would be a world record for a teenager.
Elsewhere, Sunderland signed out-of-favour Liverpool striker Borini for a fee that could rise to $22 million.
The 24-year-old Italian, who was also linked with Inter Milan, Fiorentina and Watford, returns to the Stadium of Light for a second time having enjoyed a loan spell during the 2013-14 season.
He arrived as winger Emanuele Giaccherini joined Bologna on a season-long loan.
West Brom confirmed the sale of Nigeria striker Brown Ideye to Greek club Olympiacos for an undisclosed fee.
Ideye joined the Baggies for a then club record $22 million last year, but scored just seven goals in 31 appearances.
Norwich signed DR Congo striker Dieumerci Mbokani on a season-long loan from Dynamo Kiev with an option to buy.
Zambian striker Emmanuel Mayuka left Southampton for Metz, signing a three-year contract with the French club after making only 16 appearances for the Saints in three years.
Southampton also signed a player, bringing in 17-year-old goalkeeper Harry Lewis from Shrewsbury for an undisclosed fee.
Promising Manchester City centre-back Jason Denayer, who impressed on loan at Celtic last season, has temporarily moved out again, and will play for Galatasaray this term.
Liverpool signed 18-year-old Nigerian striker Taiwo Awoniyi and immediately loaned him to German second division side FSV Frankfurt.
The club also announced that striker Samed Yesil and teenage midfielder Sergi Canos have gone out on loan, joining Swiss side FC Luzern and Brentford respectively.
