Two goals from substitute Xherdan Shaqiri condemned United to defeat on Sunday, and Mourinho's side now sit 19 points adrift of Premier League leaders Liverpool.
Mourinho left several star players, including Anthony Martial and Paul Pogba, on the bench, with the latter not brought on despite United attempting to haul themselves back into the contest late on.
However, when asked whether he had the full support of the United players, a frustrated Mourinho claimed the question suggested that his squad were dishonest.
"Are you calling the players dishonest?" Mourinho told Sky Sports .
"I believe they are honest, and you believe they are dishonest.
"A football player has to give their all every day, their maximum every day, it doesn't matter who the manager is.
"The club that pays him, the fans that are for 24 hours every day in love with the club, they need to respect that.
"If the players don't do the maximum, then you call them dishonest. I do not call the players dishonest."
29 - Manchester United have now conceded 29 goals in the Premier League this season; one more than they did in the entirety of the 2017/18 campaign (28). Update. https://t.co/mUxxEdYt2V— OptaJoe (@OptaJoe) December 16, 2018
Liverpool went ahead courtesy of Sadio Mane's 24th-minute opener, but a mistake from goalkeeper Alisson enabled Jesse Lingard to restore parity against the run of play shortly after.
Shaqiri's 70th-minute introduction from the bench quickly paid dividends for Jurgen Klopp, but Mourinho laughed off the suggestion that United had been fortunate to be in the game up until that point.
"If you want to speak about fortunate, talk about Liverpool's second and third goals," Mourinho said.
"Our goal was [Romelu] Lukaku getting behind and playing a good cross and then Lingard arriving in a good position.
"The goalkeeper made what I used to call half of a mistake. To speak about deflected goals, we have to speak about the second and the third [for Liverpool].
"They are a strong team, they showed they have the strongest team, but it's hard to concede the two goals we did in this moment, even harder the way the goals happened.
"Our difficulty in the game was the first 20 minutes, we couldn't cope with their fantastic speed, intensity and pressing. After that we were quite comfortable in the game, it was controlled."