De Gea was in Manchester on Monday for a medical, with a deal reported to be worth around £18million for the player expected to be completed soon.
The 20-year-old wrote on his official Twitter account: "I have been invited by Man U to see the club and the city with my family and to be given an offer."
De Gea - a European Championship winner with Spain's Under-21s on Saturday - is the man chosen to fill the void left by veteran Edwin van der Sar's retirement.
He would be the third new arrival at Old Trafford this summer following the signings of Phil Jones and Ashley Young earlier this month, and is set to push United manager Sir Alex Ferguson's spending in June to around £50million.
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.
He became first-choice stopper towards the end of the 2009-10 campaign under new coach Quique Sanchez Flores and played in the Europa League final victory over Fulham in Hamburg.
Ferguson revealed a month ago United had been working on a deal to sign De Gea for "quite a while" and although the youngster denied at the time an agreement had been finalised, the completion of his transfer now appears to be imminent.