Fellaini joined United from Everton in a £27.5m deadline day transfer from Everton, but reports claimed prior to the move that Arsenal were also keen on the Belgian international.
And Fellaini has now confirmed the Gunners were one of several clubs to register an interest.
"I don't want to tell you everything what happened during the transfer," he told 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."
In the frantic days and hours before the deadline both United and Arsenal were very busy. Both Premier League clubs needed reinforcements after a slow summer window.
In the end, United won the battle for Fellaini while Arsenal snapped up Mesut Özil from Real Madrid in a £42.4m deal.
And so far, Özil has been the more successful summer recruit - the German international has two goals and three assists from six Premier League appearances as the Gunners top the table after nine games.
Fellaini has been limited with a wrist injury, and has made just two Premier League starts for his new club.