United lifted their 20th top-flight crown at the end of Alex Ferguson's final season in charge in 2012-13, but fell well short in the next campaign after the appointment of David Moyes failed to work out.
Following an inconsistent start under Louis van Gaal in 2014-15, United have now strung together a run of six consecutive victories.
That sequence has seen them climb to third in the table, eight points behind leaders Chelsea, with some starting to tout United as possible title contenders.
In an interview with The Telegraph, Carrick stopped short of declaring United were certain candidates to challenge Jose Mourinho's men and second-placed Manchester City, but says a 21st Premier League crown is certainly the aim.
"I don't want to say we are going to win the league, but certainly at this point, it is what we are aiming to do," he said.
"We are not just looking to be happy for top three or top four.
"We want to be looking, over Christmas and into January, to be up there and then taking from there. But winning the league has to be the aim.
"At this stage of the season, we are exactly where I think we should be and talking to the lads, we have that mentality that we want, and expect, to be challenging.
"It's not a case of thinking that third is great because we have improved.
"We are looking past that and want to be challenging right up at the top."
