United won the Premier League 13 times between 1992 and 2013, but endured a horror 2013/14 season, finishing seventh.
The Red Devils have shown some improvement this term, currently sitting in fourth place on the league table, and Fellaini wants to see his side on top of the points table this time next year.
"I think Chelsea will win it [this year]," Fellaini told the Daily Mail. "We need to be there next season to show everyone that we can be champions again. To do that we must work hard and to improve too."
Fellaini believes that United's 2-1 win over Liverpool at Anfield on Sunday was extremely important as it opens up a five-point lead between the two teams.
As United are on track for a top four finish, it is likely that they will feature in next year's Champions League after having missed out this year, which will be a big boost for the club.
"It's massive because this season we didn't play the Champions League," Fellaini said. "So, for a team like Manchester United, we need to make sure we qualify for next season."
The Belgian added that he is enjoying playing under head coach Louis van Gaal and this is reflected in his performances, having scored five goals in 23 matches this season.
"Van Gaal is good," he said. "He has his philosophy, his own tactics. He has a vision for the team and he knows what he wants from the players, so it is good to work with him.
"I've played more games this season, so that is good personally. However, this season we haven't won a trophy, but I now hope we finish in the top four to play Champions League."