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The Blues made what was then a world-record £100 million bid for the Brazilian superstar just four months after Sheikh Mansour had bought the club, but Kaka opted to remain with AC Milan for the rest of the season before a £56.1 million switch to Real Madrid in the summer.
His stint with the Spanish giants did not go to plan, however, and, after being forced to return to Milan four seasons later, Cook insists that the soon-to-be Orlando City playmaker wishes he had taken City up on their offer when he had the chance.
'I have actually bumped into Kaka and he said: 'Sometimes I wish I had come,"' Cook told the Blue Moon podcast.
'It was an attempt to try and get one of the best players in the world to come to the football club. There is a realism to knowing you have a very, very slight chance but it was a representation of our ambition.
'It made a lot of people go: "Hold on a minute, maybe these people are serious" - which impacts players, impacts the fans, impacts the employees and the world of football is starting to look different. All of a sudden, there is a new kid on the block.'
City youngster Jose Angel Pozo, meanwhile, made the move from Real to the Etihad Stadium in early 2012, and the 18-year-old Spaniard maintains he has no regrets about his decision.
The promising striker scored a debut goal in Wednesday's 7-0 rout of Sheffield Wednesday in the Capital One Cup and believes the move to ditch the La Liga outfit is already proving a wise one.
'Manchester City gave me another vision,' he told M.E.N. Sport. 'They showed me their plans for the future and when I got the opportunity, I took it.
'It is very difficult to get into the first team at Real Madrid, as every summer they sign the best players in the world.
'There are also very good players here in the first team, but I think I took the right decision and now it is down to me and the work I do every day.'
The 36-year-old midfielder scored two goals in midweek to add to his strike on Sunday against former club Chelsea, prompting suggestions that his stay could be extended, but the Chilean boss stresses that the decision will come down to what's best for the Premier League club.
'I can clarify the situation,' he said.
'It is very easy. Lampard is playing. He has scored three goals in the last two games. In this moment, he is a very important player for us. He will stay with us here until January.
'We will see after what happens in the future. It is not a problem we have to analyse at the moment. We are here in September. There is October, November, December, three long months he continues with us.
'We will see after when the loan finishes what we are going to do. The possibility is to do the best for Manchester City.'
Pellegrini will be without Stefan Jovetic, Fernando and Samir Nasri for Saturday's Premier League clash against Hull City, but it remains to be seen whether Lampard will be given a second consecutive start, with David Silva and Sergio Aguero both passed fit for the match.