Manchester United boss Mourinho has engaged in several battles this season, with Conte and Scholes among the most high-profile.
The 54-year-old has regularly blasted Chelsea boss Conte, recently aiming digs at the Italian over match-fixing allegations made during his time as Juventus manager.
But the Sun say it is his criticism of club legend Scholes that angered Manchester United the most.
Mourinho recently launched a scathing attack on the former England international for comments made about Paul Pogba.
“Paul [Pogba] tries to do his best all the time. Sometimes he plays very well, sometimes he plays well, sometimes he doesn’t play so well,” he said after their 2-0 win over Everton on New Year’s Day.
“It’s not Paul’s fault that he has made much more money than Paul Scholes.
“It’s not Paul Scholes’ fault either, it’s just the way football is.
“I think Scholes will be in history as a phenomenal player, not as a pundit.
“I prefer to look at him as a phenomenal player that gave so much to the club that I am proud to represent.
“If Paul one day decides to be a manager, I wish that he can be 25% as successful as myself. 50% of that is 12.5 silverware, 25% is around 6.
“If he’s 25%, he’ll be quite happy. In my mind Paul Scholes is a phenomenal player, one of the best I’ve ever seen playing in midfield.
“He gave so much to my club that I can only thank him for that. The prestige of this club is based on people like him.”
The Sun say Mourinho’s comments did not help his relationship with United chiefs.
However, they still think he is the right man for the job.
The 54-year-old is likely to sign a new contract after previously being tipped to join PSG.