Diaz currently plays for Argentine second division leaders Agropecuario and, despite playing just seven league games this season, has caught the eye of a number of clubs in Europe.
However, the Daily Mail reports that Mourinho has now identified 21-year-old Diaz as a potential transfer target to solve the problem.
Speaking of the interest, Diaz told Ataque Futbolero: "I found out about it on Saturday morning, but later I did not talk to anyone other than my agent.
"My agent told me that they were delighted and they were going to see me a couple more games. It is the dream of any footballer [to play for Manchester United]."
United currently sit second in the Premier League table, but eight points adrift of runaway leaders Manchester City.