Moyes has never beaten Mourinho in nine head-to-head battles and the pair will be in opposing dug-outs again on Sunday when Manchester United head to Stamford Bridge for an eagerly-awaited encounter with Chelsea.
Intriguingly, both men were linked with the manager's job at the other's club at some point in their careers and it is widely assumed Mourinho was on United's hit-list, only for the club's hierarchy to pass over the Portuguese in favour of Moyes when Sir Alex Ferguson opted to stand down last summer.
The wisdom of that decision cannot be assessed properly until much further on.
However, with Champions League triumphs from his time at Porto and Inter Milan and the impressive record of winning championships in four European leagues, including England, Spain and Italy, Mourinho has established a reputation that goes far beyond his present employment.
And Moyes, as he made perfectly clear in the summer when he shot down the first suggestions of the pair being at loggerheads, has nothing but respect for Mourinho's work, even if he did try and lure one of United's best players to Stamford Bridge in the summer.
"I have no problems at all with Jose," said Moyes.
"We were never going to sell Wayne (Rooney), so my relationship with him has been fine.
"He has done a brilliant job wherever he has been. He is someone who catches people's imagination.
"The manager I have taken over from, Sir Alex, is undoubtedly the best of his generation and longer. Jose could well go on to be the best of his generation because he has been so successful in a lot of the jobs he has been in.
"I have only got admiration for Jose, nothing else. But I am competing against him and I am going to try and beat him every time I play against him.