Mata was sold by Mourinho during his second spell at Chelsea to United amid reports that he was unhappy with the Spaniard's lack of pace and work rate.
The duo have since been reunited at Old Trafford, with Mata featuring prominently for the Red Devils this season.
The Spain international revealed that his family and friends had feared that he could be sold by United following Mourinho's appointment as manager in the summer.
His future at the club appeared to be in doubt after he was brought on as a substitute and taken off in the space of 30 minutes by the Portuguese coach in the Community Shield win over Leicester City in August.
"There were so many things said about what was going to happen when the new manager arrived. In my mind I had it clear," Mata was quoted as saying by the Sun.
"When the manager came in I was thinking the same as always — I'm going to give my best and believe in myself and I think this can work out.
"I'm happy with the football I'm producing, that's the most important thing. I do feel an important part of this team, I do feel that.
"The truth is nothing happened [with Mourinho at Chelsea]. I came here to play for Manchester United as one more of the players in the squad. That's it," he continued.
"I started training from the first day I arrived, to get fit and be ready, and that is the case now."
Asked about his relationship with Mourinho, Mata said: "It's normal. It is a professional relationship. That's it."
United travel to Istanbul to take on Fenerbahce in the Europa League on 3 November.