The 20-year-old had a medical at Old Trafford on Monday in preparation for a deal worth about £18.9m.
He wrote on Twitter: "I have been invited by Man U to see the club and the city and to be given an offer."
De Gea was part of the Spain squad which won the European Under-21 Championship on Saturday.
A product of Atletico's youth academy, De Gea spent his first professional season with the club's reserves in the Segunda Division B before making his senior debut as a substitute in the Champions League against Porto in September 2009.
His arrival at Old Trafford follows the departure of Edwin van der Sar, who retired after the Champions League final defeat by Barcelona in May.
In May, United manager Sir Alex Ferguson confirmed that United had been working on the De Gea deal for "quite a while".
"He's a young goalkeeper, very quick, good composure, presence and an outstanding replacement for Van der Sar," Ferguson told the club website.
"We were looking for the same type of qualities as Edwin, because the one great quality Edwin always had was his composure and organisational ability."
Ferguson currently has goalkeepers Tomasz Kuszczak, Anders Lindegaard and Ben Amos on his books.
However, De Gea - assuming he accepts United's offer - is expected to be number one and former Manchester United striker Diego Forlan believes he will be a worthy successor to Van der Sar.
Forlan, 32, has played alongside De Gea at Atletico for the past two years and told BBC Sport earlier this month that the goalkeeper is a star in the making.
He said: "There's no doubt that David will become a very, very good player, because he has the talent.
"He is tall, strong on crosses, a good shot stopper and distributes the ball very well.
"I suppose he is similar to Van der Sar in a lot of ways. Even though he is very young, he gives a lot of confidence to his defenders and has a lot of talent."
The deal will take the Premier League champions' summer spending past the £50m mark after the acquisitions of defender Phil Jones from Blackburn and Ashley Young from Aston Villa.
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