United signed Matic, 29, on a three-year deal from the Blues for £40m ($52m) last month, making him the third summer arrival at Old Trafford after centre-back Victor Lindelof and striker Romelu Lukaku.
The Serbia international made his Premier League debut for the Red Devils in the 4-0 win over West Ham last weekend and was named man of the match for his disciplined performance in midfield.
When asked by Sky Sports if United are the biggest club he has played for, Matic said: "Definitely, everyone knows what Manchester United is about.
"When you say Manchester United, it is different. Also Chelsea is a big club, Benfica are the biggest club in Portugal. But Manchester is Manchester.
"When I came here for the first day, I could feel that the people at the club help you everyday if you need something and supporters are very important for this club.
"After every training session, they are waiting for us outside the training ground to take pictures, to give a signature. So I think the club is one of the biggest in the world.
"Maybe you have two or three clubs similar to United, but Manchester is one of the biggest clubs in the world for sure," he continued.
Matic added that Mourinho was definitely a major factor in his decision to leave a title-winning Chelsea team and join United.
"[Mourinho] bought me to Chelsea from Benfica and now when I decided to come to Manchester, he was a reason to come here," the Serbian midfielder said.
"Mourinho was the reason and so was the club, Manchester United. So you don't need to think a lot about if you are going to come. My decision was very quick and very clear."
United's next league assignment sees them taking on Swansea City at the Liberty Stadium on Saturday (19 August).