The 25-year-old Belgian midfielder joined United on deadline day from Everton in a £27.5million move, the only substantial deal that the Premier League champions managed over the course of the summer.
Despite seeing Arsenal in top form, and some way ahead of United at this stage - Fellaini says he does not regret his decision to move to Old Trafford.
"I don't want to tell you everything what happened during the transfer," he said to beIN Sport. "This was done at the last minute.
"In a transfer window, there is always a list of players, you never know if it's true or not. But for me, I knew [Moyes] was interested in me.
"Chelsea? No comment. Arsenal? It's true, they were interested, but Manchester United convinced me to sign for them."
Fellaini says he could not turn down the chance to work with David Moyes again.
"David Moyes has given me confidence," he said.
"When I was in trouble early at Everton, he helped me and supported me. He did everything for me to become a great player.
"I was criticised, nobody knew who I was. I succeeded because I worked a lot. That's why I'm playing for Manchester United today.
"[I am not trying to] replace Scholes or Carrick or anyone. I try to play my own game. It is difficult at the moment, but I will succeed."