The Belgian moved to Goodison Park for £15million hours before the transfer window closed on deadline day.
The deal raised a few eyebrows and some critics have argued Moyes was simply desperate to bring in another new face after a largely frustrating summer.
Moyes accepts he was looking at a number of other players but insists Fellaini was always in his thoughts as well.
"People will say that Fellaini was a last-minute thing and he wasn't on the radar and a lot of that is correct," Moyes told Everton TV.
"We were after other people; [Stephane] Mbia, we were after two or three people who we couldn't get.
"But we weren't sitting about doing nothing. We were looking to see who our next targets were."
He explained: "We haven't got a lot of stature in the team, so our criteria was that we had to get someone who was bigger than what we had.
"We have a lot of smaller people in the team, Arteta, Osman, Pienaar, so there were lots of good midfielders about but they didn't meet the criteria.
"Fellaini was one who we knew could do that and we have watched him.
"His games against Liverpool were very good and that probably tipped us over. We realised then that instead of having the potential, he was there."