Louis van Gaal launched a €20 million [£16m] bid for the Argentina defender earlier this week, but United saw their approach rejected by the Portuguese outfit.
That prompted Rojo to reportedly miss training on Wednesday in a bid to force the move, but 24-year-old has now apologised for his actions and admitted to acting rashly following the failed bid.
"It was all a misunderstanding, I was a hothead. I'm sorry for what happened because I feel that I was unprofessional," he told the club's official TV station.
"My career depends on Sporting. Now I will try to correct the situation. At the time Sporting signed me, it was not a good time. I can only thank Sporting.
"I want to tell the people that I didn’t behave correctly, but I’m am very fond of this club. They are the ones that will decide my carreer, and I hope it will be for the best both for me and the club. But I want to stress that I’m very happy at the club, and I like the way I’m treated and the care they give me."Following United's shock 2-1 loss to Swansea City on the opening day of the season, Louis van Gaal insisted he is not concerned by the club's lack of signings despite bringing in just Ander Herrera and Luke Shaw.
Goal understands that United are also pushing hard to sign Juan Cuadrado and Daley Blind and has called for patience following his appointment as he looks to make his mark on the squad.