The Red Devils, who the striker also skippers, secured the final Champions League place last season after a slow start to the campaign, but they still finished 17 points behind champions Chelsea.
Louis van Gaal's squad has already been boosted this summer by the arrival of Memphis Depay and Rooney is confident that United can compete at the very top with the right additions to the squad.
"I honestly believe we will go far next year and won't be far off challenging for the league," he said on his website.
'It would be good to win as many cups as possible, a club like Manchester United should be winning trophies'
Wayne Rooney
"If you look at the first 10 to 11 games we only had around 13 points, and if you look at where we finished I think it shows we could have been up there with a better start.
"We've dominated and outplayed most teams this season but haven't taken our chances to win those games, it could have been a different story if we had.
"The most important thing is that as a squad, including the manager and myself, we believe we are close to challenging for the title and with a couple of signings in the summer we will have a good chance of doing that.
"It would be good to win as many cups as possible, a club like Manchester United should be winning trophies and hopefully next season we can put that right."
The 29-year-old was also full of praise for Van Gaal in his first season in English football after replacing David Moyes and, while he was disappointed not to win any silverware, Rooney was philosophical about the campaign and glad the club got "back where it belongs".
Rooney scored for England against Slovenia earlier this month
"You have to realise that at the start of the season we had a new manager, new coaches and a lot of new players, so it was always going to be a tough ask for us to go and win the Premier League," he added.
"It was a bit disappointing not to win any silverware, especially the FA Cup after the defeat to Arsenal in the quarter-finals, we had a great chance of going on to win that.
"After last season it would have been a disaster if we didn't get that Champions League spot, but the main thing is that the club is now back where it belongs.
