But, according to the London Evening Standard, the Stamford Bridge chief expects United to keep the faith with Moyes.
The defending Premier League champions have struggled under their new boss, and currently sit seventh in the Premier League table.
Despite the difficulties, Mourinho believes United will afford Moyes plenty of time to get it right on the pitch.
"My feeling, which is based on years of communicating with Sir Alex and some inside information, is Man United are not happy, but they are calm," Mourinho told the Evening Standard.
"They trust David. David trusts them. I don't think David is under pressure. The pressure is something virtual. It is something that comes from the media as a consequence of not having the best results.
"The most important thing in all of this is not the pressure that comes from the outside, it's the reality of the inside.
"The manager is calm and they are thinking this is our manager for the next two, three, four, five years. So they're all together and are going to rebuild again a big team.
"The bad results will finish. They will get back on track [although it is] probably too late to win the league because they're too many points behind the leader."
Some sections of the media claim Moyes is already under pressure after a disastrous run of results at Old Trafford that included defeats to Tottenham, Everton and Swansea.
In contrast, other new managers at Manchester City - and Mourinho at Chelsea - have fared much better with the two Premier League giants still well in the title mix.
Having said that, neither had to succeed Sir Alex Ferguson - the most successful British manager of all-time.