Argentina international Rojo signed for United from Sporting Lisbon last month in a deal worth £16 million, but has not yet featured for his new club due to delays in him being given a visa.
However, the UK Border Agency granted the left-sided defender permission to live and work in the country earlier in the week, meaning he is now free to play for United.
Rojo is therefore in line to make his Premier League bow when Louis van Gaal's side host QPR after the international break on September 14.
And the 24-year-old, who was impressive in his country's run to this year's World Cup final, cannot wait to get going at Old Trafford.
"I am desperate to get onto the field and play because I have not been able to step into Old Trafford yet," Rojo told the Argentinian media.
"I am very happy to be at Manchester United, a top team in the world, sharing a dressing room with players like Wayne Rooney, Robin van Persie [and fellow new signings] Angel di Maria and now Radamel Falcao.
"This is a very important step in my career as a player and as a person.
"Everything that has happened to me since the World Cup has been a dream."
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