United stumbled to their sixth and perhaps most embarrassing home defeat of the season on Tuesday when they went down 3-0 to bitter rivals Manchester City.
The last time United lost so many home games in the league was in 1978.
Anger followed the defeat, with one fan having to be escorted out of the south stand after approaching the dugout to vent his anger against the United manager.
There were reports of fans aiming abuse at Moyes' predecessor Sir Alex Ferguson, who was sat in the directors' box.
Ferguson hand-picked Moyes from a strong field last summer.
For that reason, a banner has hung in the Stretford End that includes Moyes' face with the words 'The Chosen One' written alongside it.
Some fans have called for the group that made the banner - Stretford End Flags (SEF) - to remove it following the 3-0 defeat to Liverpool almost two weeks ago.
And more fans called for its removal on Tuesday in the wake of a third successive home derby defeat.
Stewards also gathered around the banner at full-time to prevent it from being seized.
It is understood that the SEF has no intention of taking the banner down ahead of Saturday's Barclays Premier League match against Aston Villa despite calls to do so on Twitter.
Although there were few - if any - boos from the home supporters who were left at the final whistle on Tuesday night, there was a noticeable lack of songs in support of Moyes.
The Manchester United Supporters Trust (MUST) thinks it is understandable that the club's fans are disappointed by the fact that their team now sit seventh in the table, some 18 points adrift of leaders Chelsea.
"If your ego has been continuously set on success for 20 years and then to suddenly you do not have it, it's obviously going to affect you emotionally in a difficult way," MUST spokesperson Sean Bones said.
MUST thinks United's troubles do not lie solely at the feet of their manager, however.
It believes a lack of investment from the Glazer family since their controversial takeover in 2005 has also been a key factor in United's failures this term.
It also believes the Glazers' refusal to go toe to toe with the likes of City in the transfer market has left United short on quality compared to their 'noisy' neighbours, who are now favourites for the title.
"We think the problems lie further deeper than David Moyes. They sit at the table of the Glazers," Bones said.
"The Glazers have not invested correctly at the critical times.
"The Glazers have had an element of success, but we feel that success was based on the investment made by the PLC in players like (Wayne) Rooney, (Cristiano) Ronaldo and Rio Ferdinand - the very best possible players and we invested in the Class of '92 and brought them through and the Glazers got the benefit of that.
"And the problem is when you don't invest correctly at the cortical points, you get yourself into the position where there is a huge chasm of class between the ones who have invested and the ones who haven't and it costs a lot more to repair the damage.
"Sometimes it can take a long time to repair it."
MUST also thinks the Glazers made a mistake by allowing chief executive David Gill to leave the club at the same time as Ferguson.
"Last summer we had a new manager and a new CEO in Ed Woodward, he is unproven in that job as well, so it was a bad decision to allow that to happen," he said.
"They could have held on to David Gill to try to assist the smooth running of the club."
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- Sir Alex Ferguson
- David Moyes