Incoming board chairman Colin Graves promised in March there would be serious questions asked should England not beat a team he termed "mediocre".
While Cook showed a promising return to form with the bat, striking a century in the third test loss in Barbados, Moores may feel the heat after the series ended a disappointing 1-1.
In his second stint as coach, Moores has had his share of inquisitions and was already under scrutiny after England's group-stage exit from the World Cup.
That scrutiny intensified with the departure of managing director Paul Downton, who appointed him last year.
Cook said Moores had brought "a lot of good stuff" but admitted decisions about his own future and that of the coach were out of their hands.
"I don't think it's inquiry time when you look at how we played over the 13 days, but I think we can all sit in that dressing room and know we've let an opportunity go," Cook said in comments published by British media.
Pundits and former players were less sanguine.
"How much Moores?" asked British tabloid, The Sun, in a headline on its website.
"Sometimes you have to accept it's not working," former England captain Michael Vaughan tweeted.
If hauled before Graves, Moores will have to defend his team on a number of fronts. Some issues he can claim were beyond his control.
England wilted under pressure, said Cook, perhaps an understandable reaction from a young and still-developing team.
Yet batsman Jonathan Trott's reinstatement will be harder to justify after the veteran's poor series.
Selectors are also under scrutiny for not picking another spinner after part-timers Moeen Ali and Joe Root struggled.
While the England board are quiet on whether there is space for a coach under the new more powerful 'director of cricket' role to be filled, questions about Moores' future ahead of home test series against New Zealand and Australia will continue.
Highly-rated former Australia test bowler Jason Gillespie, coach of county side Yorkshire, has been touted as a potential successor.
Shortly after England's loss in Barbados, Gillespie told the Yorkshire Post he had not applied to become a full-time coach his home state South Australia in the domestic Sheffield Shield competition, raising speculation he could be set to take the England reins.

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