United defeated fierce rivals Manchester City 4-2 on Sunday afternoon to move above them in the Premier League table, and just a point off Arsenal in second place.
While Chelsea look set to win the title, a second-placed finish is very much within their grasp and would finish off what has been a fine run of form at the end of the season under Dutchman Louis van Gaal.
Now, Carragher has said that United are headed for the title under the former Netherlands manager’s leadership.
“Looking at them now, people will start getting the idea they could go on and win the league next season,” he said. “People have more belief now in what he’s trying to do.
"Next season he will be bedded in, the football will come, he’ll have more players he wants and people will be more used to him.”
United suffered an indifferent start to the season as Van Gaal failed to find a system that consistently worked for his side, notably playing Wayne Rooney in an unfamiliar midfield position.
They also came in for criticism over their tactics and style of play, having been accused of long-ball football by West Ham United manager Sam Allardyce, according to Sky Sports.
They were, however, still winning games and maintaining their push for the Champions League, and Carragher says that is what was most important.
“When you finish seventh last year and a new manager comes in, your job is to get into the top four,” he added.
“It doesn’t matter how they play, or were playing, they were getting results. United have been in the top four for the last three or four months now probably, and it’s only in the last four to six weeks that we’ve seen the performances to go with it.”