The Washington Post says the 32-year-old wants to make the move to the United States.
The boyhood Blue only returned to Goodison Park last summer.
United officials said they did not want to discuss Rooney's visit, but according to several people with firsthand knowledge, both sides used the trip to get to know one another better.
“He wants to come," said one person, who met Rooney here. “He wanted to make sure D.C. was serious, and he wanted to get comfortable with the idea of playing here."