Frenchman Clichy, 25, joined City from Premier League rivals Arsenal on Monday, ending an eight-year association with the Gunners.
"I understand that the people who live in Manchester are the true fans. They are from City," Clichy told Manchester City's official website.
"That is a good point. I won't be bothered by United fans."
Clichy left Arsenal after the North London side failed to end a six-year wait for a major trophy, with the Emirates Stadium side crashing out of three cup competitions in the space of three weeks in February and March, before sliding out of the title race and eventually finishing the season in fourth place.
And the full-back insists there is no limit to what the Eastlands side can achieve, and claims the main difference between City and Arsenal is the strength of the respective playing squads.
"Of course, you have to be realistic but there is no limit at Manchester City. Where they have come from in three years is phenomenal. And I want to win things," Clichy said.
"My season was a bit like the team's. We started well and collapsed at the end of the season after the Carling Cup final.
"These things happen in football. That is why you can see the importance of having a big squad.
"Sometimes it is difficult to have a lot of players because you have to leave some out. But you need to have a big squad to go all the way. That is why I joined City."
Despite appearing to goad Manchester United's fans, Clichy was full of praise for his compatriot and fellow left-back Patrice Evra, but stated that he hoped success with his new club would help him usurp the Red Devils full-back at international level.
"Patrice is a great player who plays for a great club. You have to accept this," said Clichy.
"Now, I hope it will be a different story. I am at a different club.
"Hopefully my team will be the winner and that will take me higher with France."
Clichy won't have to wait long before going head-to-head with Evra and Manchester United, with City set to face their arch-rivals in the Community Shield at Wembley next month.
