"This football club is, without doubt, going to be the biggest and best football club in the world," Cook told a fans forum in New York.
"I will make no excuses for saying it, as I truly believe it - with the resources and capabilities we have."
Cook was speaking before the Football Association urged both Manchester clubs to show restraint ahead of their Carling Cup semi-final second leg next Wednesday after City won the first leg 2-1.
In a tense game former United striker Carlos Tevez's goal celebrations prompted a one-fingered gesture from Gary Neville, while subsequently the Argentine branded his ex-United team-mate a "moron" and a "foot-licker".
Since Mansour took over in 2008, City have spent almost £200m on players as they look to break Manchester United, Chelsea, Liverpool and Arsenal's stranglehold on the Premier League's top-four places.
Can Manchester City dominate the world?
City's spending power has also eclipsed their neighbours, with United's holding company debt growing to more than £700m, though it did announce on Friday that a bond issue had raised £504m.
Last summer Sheikh Mansour's financial muscle ensured City brought Tevez, Gareth Barry, Emmanuel Adebayor and Joleon Lescott, while United opted not to spend the bulk of the £80m brought in by Cristiano Ronaldo's sale.
City also angered United in the summer by displaying a "Welcome to Manchester" poster on completing the transfer of Tevez from his advisors, the Argentine striker having been on loan at United who deemed an asking price of £25m to be excessive.
United manager Sir Alex Ferguson retorted by branding City a "small club" while later describing them as "noisy neighbours" after his side's controversial 4-3 derby victory in September.
"It's City isn't it? They're a small club with a small mentality. All they can talk about is Manchester United; they can't get away from it," added Ferguson.
"They think taking Carlos Tevez away from Manchester United is a triumph. It is poor stuff."
Cook's comments follow on from those he made at the club's tour of South Africa in the summer where he said he hoped Manchester City would become "the darling of world football".
After Tevez moved to City he explained that the club's vision and "ambition to become the biggest club in the world," had helped convince him to join.
The 25-year-old scored just once in his first six games for City, but has recently hit 13 goals in 11 games, including both efforts in City's midweek Carling Cup success.
Source: BBC Sport
Source: BBC Sport