Mourinho has made United striker Wayne Rooney his number one transfer target this off-season, with Chelsea already having a bid rejected.
But Moyes, under pressure to deliver at Old Trafford, spoke glowingly about Mourinho and said there was no conflict.
"Jose is a tremendous manager. He has done really well in the Premier League. When he lost his job at Chelsea last time I wrote him a letter," the Scot said.
"So, for people to say there is any feud is wrong.
"Of course, you are going to find two really competitive managers.
"When our teams play we will both be highly committed to make sure our clubs win, but I admire him for how well he has done as a manager."
Moyes is eager to bring Barcelona midfielder Cesc Fabregas to United, but the former Arsenal captain appears unlikely to leave the La Liga champions.
Despite United's poor pre-season results on their Asian tour, Moyes believes United have the squad to defend their English Premier League title – with or without Fabregas.
"This is the squad that finished 11 points clear at the top of the league last season so, yes, I'm confident," he said.
"What we have is a really good squad — you can't forget about the quality that's already here.
"We're hoping to add to it and we're confident we will do before the window shuts. We've got until September 2 so I'm confident we'll have players in.
"But even without additions this squad will be tough opponents for any team — they will be looking at Manchester United.
"We know we will be challenged by everybody, every game we play."
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