The People says they were at loggerheads after a stormy end-of-season summit with the club's Arab owners in Abu Dhabi when they clashed over their views of how the club is run.
City were notable absentees from the transfer market at the start of business as Mancini wanted things run his way while Cook and his sidekick, football administrator, Brian Marwood stood their ground.
But Cook and Mancini last week went through their differences.
A source said: "Roberto wanted more players and more staff last January but he did not get the help he wanted.
"He knows if he had not delivered he would have been sacked but the FA Cup and reaching the Champions League changed that."