The Blues will have the opportunity to do the double over arch rivals Manchester United when the teams meet at Old Trafford this evening.
Pellegrini has warned that City must progress further if they are to establish themselves as a major force.
"I think this club has been trying to work in the last four years to reduce the difference we have against important teams - against Manchester United, Chelsea and Arsenal, and in Europe as well," he told Sky Sports News.
"It is always important because when you have two important teams from the same city you always want to be the best team.
"But we are not just fighting against Manchester United and in some ways it doesn't matter."
City will move to within three points of league leaders Chelsea if they win at Old Trafford.