A touchline clash between managers Roberto Mancini and David Moyes towards the end of Everton's 2-0 victory at Eastlands on Wednesday, which resulted in both men being sent to the stands, highlighted the simmering tensions between the clubs.
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The FA has received referee Peter Walton's match report and the disciplinary unit is now due to study it before deciding on possible further action.
Mancini apologised personally to Moyes and Walton following the game.
The bad blood now appears to have spread to the boardroom, with Everton contacting Cook in the wake of an alleged row with George Downing, a guest of an Everton director at the game.
After Mikel Arteta scored an 85th-minute goal to confirm Everton's victory, it is alleged that a remark was made to Cook in reference to a chant from the visiting supporters of "2-0, and we've spent ---- all", a jibe aimed at highlighting Everton's success at building a team on a tight budget in comparison to City's £120 million summer spree.
Cook and Downing, a regular guest of the Everton board, are alleged to have exchanged words in the directors' box before the latter, who was attending the game with his wife, was asked to leave.
While City have stressed that Downing was not escorted out of the stadium by security, it is understood that he left of his own accord after being asked to do so.
With Downing, who has made his fortune in the construction industry, attending as a guest of the club, Everton contacted Cook on his behalf on Thursday before issuing the following statement: "We have written to Mr Cook asking for his observations with regard to the incident."
Downing is understood to want an apology from City.
City were privately bemused by the furore that has been created by what senior figures have suggested to be nothing more than Cook being unhappy with what he thought was behaviour not usually associated with the directors' box. It has also been claimed by City that Cook quickly brushed aside the incident before sharing a post-match drink with the Everton chairman, Bill Kenwright.
The controversy surrounding the Lescott transfer continues to rankle with Everton, some of whose officials believe that the club were treated with a lack of respect by City.
Everton manager Moyes claimed at the time that "the way it [the Lescott deal] has been handled is disgusting and all it has done is disrupted our club. It has not been handled correctly from over there [City]". The response from Mark Hughes, the then City manager, claiming that he was not surprised that Moyes felt a "bit aggrieved with the situation", only fuelled the anger towards City within Goodison Park.
Lescott has missed both games against Everton both won by his former club but Moyes alluded to the row by admitting after Wednesday's victory that it was important for his club to show "class and dignity" at the City of Manchester Stadium.
Cook's alleged involvement in the spat is only likely to intensify the focus on his role at City, despite the support he has from senior figures close to the club's owner, Sheikh Mansour bin Zayed al Nahyan.
The former Nike executive was subjected to heavy criticism for his part in the sacking of Hughes last December and he has also been ridiculed for outspoken comments in the past, including labelling AC Milan as "bottlers" following the collapse of City's attempts to sign Brazilian forward Kaka in January 2009.
Sheikh Mansour and chairman Khaldoon al-Mubarak are understood to view Cook's input in restructuring the club as key, however, and his position internally is regarded as unquestioned.
Downing a key Everton player
George Downing is Liverpool's largest commercial landlord, who from humble beginnings in student housing has built up a massive portfolio in the last 20 years, specialising in transforming derelict or run-down buildings.
He was voted Merseyside's Businessperson of the year in 2007 in recognition of his investment in the commercial sector.
His companies own a major part of the city's office space, including the Port of Liverpool Building, the Capital Building on the Pier Head and the Chamber of Commerce's HQ.
Despite his success, he has always shied away from publicity, saying last year: 'I keep a low profile and have never sought publicity for what we've achieved.'