Moyes' one and only term in charge at United saw him sacked before the season was out, as the 20-time English champions slumped to a seventh-place finish in 2013-14.
The former Everton man refuted the suggestion he wanted out after the man who took him to Manchester from Goodison Park departed the club in April, with Moyes coincidentally appointed Real Sociedad boss on Monday.
Fellaini was strongly linked with a move to Napoli, after coach Rafa Benitez expressed interest.
"No. It was all speculation. In football it is like this," the Belgian said.
"That's what happens there is always speculation.
"But I signed for Manchester United for four years and the manager never said to me that I had to go so I worked for my place and that was it.
"When you are playing you are always happy. So I am happy at the moment."
Fellaini said the pressure of United's lacklustre season, which saw them miss out on the UEFA Champions League for the first time since 1995-96, left him feeling like a scapegoat, after arriving at the club with Moyes.
"Yes [I felt] a little bit [of a scapegoat]. That is a difficult question. It is difficult to answer that," he said.
On his faith in his own ability, Fellaini added: "That is why David Moyes bought me. He knew my qualities. He knew what I can do and that is why I am here.
"I agree last season was tough. There was a new manager and it was difficult for all the players but we have another new manager now with a new philosophy and we have to work hard."
Fellaini said he was well-supported throughout his first season at United, and predicted his form is on an upward incline after overcoming an ankle problem.
"My family helped me through it [the season]. I was also in the national team for Belgium at the World Cup and I played well and the team also played well. I came back after the summer confident.
"The manager [Louis van Gaal] said to me I would have to fight if I wanted to play. I did fight and trained hard and that was it.
"Last season was a difficult season for me. I am fit now and that is why I am better.
"When I am super fit I will be better I am sure. I was injured earlier this season. It was a bad injury but now I am better and I can look forward."
Fellaini said Van Gaal wanted him to look to score, as he occupied the spine of United's formation.
"At the moment the manager is using me as a box-to-box player and I am trying to score a goal," he said.
"I played a little bit more up front as a number eight. Then for the last 20 minutes [against Crystal Palace on Saturday] I played at number six alongside Daley Blind. I can play different positions."
