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The Argentine defender was confirmed as a United player earlier in August in a move that saw winger Nani rejoin Sporting Lisbon on loan until the end of the season.
Rojo has been unable to make his debut for the Premier League club as yet due to red tape, with passport issues dragging the process out.
"Rojo is working on his permit because he has to go abroad for that," Van Gaal explained to the press. "The difference between [fellow new signing] Angel Di Maria and Rojo is that he has an Italian passport and Rojo does not.
"He has an Argentine passport and he has to work for it. Now he is abroad in Madrid to work at the embassy but it takes more time than we expected."
Di Maria, a British-record signing from Real Madrid and a fellow Argentine, is poised to make his United debut at Burnley on Saturday.