The pair worked together for three years at Barcelona before Mourinho's Inter Milan beat van Gaal's Bayern in the 2010 Champions League final.
And they will go head-to-head next season, with United expected to challenge again after a poor campaign last term in which they finished seventh under David Moyes.
Mourinho told the Mail on Sunday: "I'm not worried about him being here. I look forward to playing against United. Nothing worries me.
"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 on August 16, two days before Chelsea open their campaign against promoted Burnley.