The Red Devils have won six and drawn one of their seven matches since Carrick's arrival as head coach as he replaced former manager Ruben Amorim.
United trail Arsenal by 13 points with a game in hand and have 10 matches to play as the final stage of the season begins for clubs aiming to lift the title and jump into the European spots.
Carrick had been brought in as an interim head coach and could become permanent manager if he secures Champions League football.
However, when asked if his side are in the title race he admitted that he still has larger ambitions.
"You can't rule anything out in football, but we've got to be realistic and know where it's at.
"I think we've just got to try and keep winning games and see. Above us now there are two fantastic teams. There's some really good teams in and around us. We've had a good run, we're certainly not getting carried away. Of course, I'm being realistic about it. We have to win a lot of football matches for that to happen.
“We won't get carried away, you've got to feed off the confidence and understand that runs of wins are hard to come by. We're hungry for more, and we need to try and make it last as long as it can and stay consistent."
United only have Premier League football to focus on for the remainder of the season as they face Newcastle United on Wednesday as they aim to close the gap at the top.