Casemiro, 32, endured a torrid 45 minutes against Arne Slot's visiting side, during which time he made two mistakes that contributed to Luis Diaz's two opening strikes.
United boss Erik ten Hag hooked the Real Madrid legend at half-time - replacing him with debutant youngster Toby Collyer - and cut a sullen figure as he left Old Trafford.
Casemiro has faced criticism after an initial standout season was followed by a struggling second season, with his Liverpool performance another difficult showing from the veteran.
Comments have been aired about Casemiro's ability to continue performing at the top physical level that Premier League football demands, but nobody inside the United dressing room has blamed him for the defeat, as reported by The Sun.
Club captain Bruno Fernandes gave the squad a half-time dressing down as Liverpool flexed their muscles at the home of their rivals, with the United skipper insisting every player had to take responsibility.
Casemiro has so far started each of United's three Premier League matches in a pivot midfield alongside Kobbie Mainoo, but faces increased competition from new signing Manuel Ugarte.
The five-times Champions League winner was previously linked with a move to Saudi Arabia after struggling in United's midfield last season, but has been kept by Ten Hag in a bid to increase his side's squad depth.
Scott McTominay's departure for Seria A giants Napoli placed even greater importance on Casemiro staying for another season at the 13-time Premier League champions.
Casemiro's future is likely to depend on his performances in the run-up to the January transfer window and a source told The Sun that: 'Nobody was pointing the finger at Casemiro after the game.
'If anything the opposite was true and the mood among the dressing room was that everybody had to improve and do better.'
Meanwhile, United legend Rio Ferdinand has also jumped to the defence of the midfielder and hit back at previous claims made by Jamie Carragher insisting that Casemiro should 'leave football'.
