Moyes's team host City on Tuesday, with most making the title-chasing visitors favourites for the Manchester derby at Old Trafford for the first time.
The Scotsman has endured a difficult first season in charge at United but argues his team are not far off a return to trophy-winning ways despite languishing in seventh place in the league, 12 points behind City having played two games more.
"We're not as far away as many people would have us and I've no doubt it'll improve," Moyes told a news conference.
"We want to show we're still in there fighting."
"In the last couple of games, a lot of the players have stepped up to the plate and performed. But I'm not surprised by that at all - that's what they're capable of and I've been saying that all year.
"We have a good squad with good players - we maybe haven't played as well as we should have, but I think we have those capabilities within the squad."
United have enjoyed a resurgence in the last week, beating Greek champions Olympiakos Piraeus 3-0 at home to qualify for the last eight of the Champions League and enjoying a comfortable 2-0 win at West Ham United in the league on Saturday, to ease the pressure on beleaguered Moyes.
However, it is in the games against the Premier League's top four teams that Moyes has endured his most torrid time, with a resounding 3-0 defeat at league leaders Chelsea, a humiliating 3-0 home loss to bitter rivals Liverpool and a 4-1 thrashing in the reverse fixture with City earlier in the season.
"We want to perform better in the bigger games than we have done, I have no doubt about that," Moyes said.
"But we don't go into it any differently - we want to win them all.
"We go out to win every game - not just the derby. Its no different. Obviously it means a lot more to the supporters and the city to win a derby game, but we go into every game with the same intention to win."
United will have to cope without suspended captain Nemanja Vidic and injured defenders Chris Smalling and Johnny Evans for City's visit, with Moyes hopeful he can recall Rio Ferdinand and Patrice Evra, who both missed the trip to West Ham.
In the absence of Vidic, who is set to leave for Inter Milan at the end of the season, and vice-captain Evra it was England striker Wayne Rooney who stepped in to captain the side in their last league outing.
Rooney scored both of United goals in the win and moved to third in the club's all-time goalscorers list, strengthening his claim to wear the armband permanently next season and possibly again on Tuesday.
"Wayne has led by example all season," said Moyes.
"But maybe he's taken a wee bit of pride when wearing that armband and showing exactly what it means. He wants to take responsibility and I think you always need players who do that."
RELUCTANT FAVOURITES Despite a reversal of fortunes for the Manchester clubs this season City boss Pellegrini moved to play down his team's favourites tag ahead of the derby.
"We are in a good moment but we are playing against a big team in their own stadium, so I don't think we are the favourites," Pellegrini told a news conference.
"Of course we always expect to go and win every game but, I repeat, just because Manchester United have had a difficult season it does not mean we are going to win. We have to go there and play a very good game."
In his first season in English football Pellegrini has led his side to the League Cup and to third place in the league, six points off leaders Chelsea but with three games in hand.
City had been touted as potential winners of an unprecedented quadruple of trophies but exits in the Champions League and FA Cup in the same week meant that the Premier League remains the one trophy they can still win.
"When you are eliminated from other competitions it's not a good thing but one positive is you can just focus on the league so we don't have problems physically or mentally," Pellegrini explained.
"We are just trying to fight for the title. I know we must go game by game and we just have to win on Tuesday and then at Arsenal on Saturday after that."
After City, United play host to Aston Villa in the league on Saturday before welcoming Champions League holders Bayern Munich in the quarter-finals on April. 1.