The Argentina striker expressed his dissatisfaction with life at Old Trafford earlier this week, bemoaning the fact he has often been left out of the starting XI in big games this season.
Fellow striker Dimitar Berbatov's arrival last summer has pushed Tevez down the pecking order - to the dismay of some United fans - and the former West Ham frontman claims he has ''many offers'' to go elsewhere.
But midfielder Carrick, for one, hopes he stays with the Red Devils to fight for his place.
''We want him to stay, Carlos is a top player. He has scored some very important goals for us over the last few years,'' Carrick told The Sun. ''We've got a world-class squad, we've got world-class players and Carlos is one of them.
''We want to keep moving forward as a team and as a squad so we'd like to keep our best players here.
''But it's down to the manager. I'm sure he knows the best way to make sure we do keep progressing.''
Meanwhile, news in The Times claims that Liverpool boss Rafael Benitez is keen to launch a raid for the striker in the summer.
It would be an audacious bid, given the two clubs' situations, but Benitez has apparently signalled Liverpool's willingness to join Chelsea, Manchester City and Inter Milan in the mêlée for his signature, with a striker high on his priority list.
Tevez would be likely to cost any other club more than the £22m that United have to pay at the end of the season to turn the player's loan into a permanent deal; but the Spanish boss is keen to make him part of Anfield's future, having failed to impress with his previous striking signing, £20.3m worth of Robbie Keane.