A poor campaign for United finishes on Sunday with a trip to Southampton, with interim player-manager Ryan Giggs' men certain to miss out on European qualification and likely to end up seventh.
Dutchman Louis van Gaal is expected to take over ahead of next season and Carrick said United were eager to contend for the Premier League title.
"There is no way we are settling for top four - that is not something we do here," Carrick told the club's website.
"We can't celebrate getting into the top four, that's not worth celebrating. We want to be challenging right at the top.
"I am not standing here and saying we are going to win the league next season but that is what our goal is and anything less than that is not worth thinking about.
"Certainly, top four is not something that we will settle for."
Moyes blamed an 'ageing' United squad for their struggles before the Scot was sacked in April after just 10 months in charge.
It has led to suggestions there could be huge changes to the squad in the off-season but Carrick believes there is still enough quality not to require such drastic moves.
"It is easy to say there will be wholesale changes because we obviously haven't had a good season," the 32-year-old said.
"If you look within the squad, there are some good players there.
"There probably needs to be some changes in some way because that is natural with the situation we are in at the moment, but you can't change too much because it is hard.
"Bringing players in is tough enough and we will just have to wait and see how it pans out."
