However, the Portuguese is not concerned about their impending rivalry in English football.
The pair worked together for three years at Barcelona but will go head-to-head this season, with Van Gaal having taken the reins at Old Trafford after a relatively successful World Cup campaign with the Netherlands.
The Dutchman has also won titles in his homeland, Spain and Germany with Barcelona, Ajax and Bayern Munich and acknowledged at his first United press conference that he is aiming to add England to that list.
Mourinho - whose Internazionale side beat Van Gaal's Bayern in the 2010 Champions League final - and Chelsea are expected to be among the main rivals for the Barclays Premier League title.
"I'm not worried about him being here. I look forward to playing against United. Nothing worries me," said Mourinho yesterday. "I watched him at his press conference and he was pure Van Gaal; honest and open, the way he always is.
"He is a very, very good coach, he'll be great for Manchester United. We are both great coaches. It's what we were born to do. Louis was the best coach in the World Cup and he is a good friend - really, a friend for life. I have deep respect for this man.
"He is in the best period of his career. With coaching, if you deal with winning and losing in the right way, you improve. He analyses every experience. He has got better and better."
Van Gaal begins his Premier League career against Swansea City on August 16, two days before Chelsea open their campaign against promoted Burnley. Van Gaal is in the United States with his new squad at the moment as they prepare for the coming campaign and the Dutch coach is expected to name compatriot Robin van Persie as the new club captain.
Identifying the player to wear the armband has been one of Van Gaal's priorities since Nemanja Vidic moved on from Old Trafford earlier in the summer. Having spent 10 years at the club, Wayne Rooney was also tipped to take the armband, however Van Persie captained the Dutch national team at the World Cup and enjoys a good relationship with Van Gaal as well.
Bryan Robson is convinced that the manager will elect to go Dutch and name his main striker as the club captain. "Robin was his captain of Holland, so that is an obvious one," said the former United captain. "I would be really surprised if he didn't give Robin it."