The Premiership champions are set to recruit Brazilian midfielder Anderson, 19, and Portuguese winger Nani, 20, in a deal that could cost about £30m.
"Terms are agreed with both players and an announcement will take place once a medical and other procedures are done," said a statement on United's website.
Owen Hargreaves will join United on 1 July from Bayern Munich for £17m.
Anderson has described the proposed move to Old Trafford as " a dream come true".
He said: "I cannot wait to play in the same team as Cristiano Ronaldo, who is a great player, as well as Wayne Rooney and many more.
"I could not imagine I would ever get to work with a manager like Sir Alex Ferguson but it has happened. What can I say? It is a dream the size of the world."
Anderson has been named in Brazil coach Dunga's squad for the Copa America in Venezuela next month but not Friday's friendly with England at Wembley.
The teenager, whose full name is Anderson Luis de Abreu Oliveira, started nine matches for Porto this season, scoring twice, and came on as a substitute six times.
He joined Porto from Gremio in December 2005 and played against United in a pre-season friendly last August at the Amsterdam Tournament.
Nani had also been linked with a move to Tottenham, but recently said he would only leave Portugal to play for a club like United.
Nani, whose full name is Luis Carlos Almeida da Cunha, was born in the Cape Verde islands and has been compared to his compatriot Cristiano Ronaldo, with pace and trickery similar to that of the Footballer of the Year.
He has played five times for Portugal, scoring once in the process. This season he featured 29 times for Sporting Lisbon, scoring five goals and he will be viewed as the long-term successor to Ryan Giggs.
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