"We know the manager has everyone's confidence anyway. No one doubts anyone's ability in the changing room or staff," Cook told Reuters in an interview.
"I don’t think that should even be a question for the chairman to ask at the moment, to be honest. We’re sitting 14th in the Premier League."
However, Bournemouth are now just six points above the relegation zone after Everton thrashed them 6-3 last weekend.
Cook, 25, said putting that result behind them in time for the arrival of Manchester City on Monday had proved a challenge.
"It’s difficult to shake off the defeat and mentally prepare for the game against City. But the beauty of football is there is always another game around the corner.
"We obviously know how tough it is going to be on Monday but it is a challenge we are looking forward to and a game we feel that, here, at home under the lights, we can do well in."
Everton striker Romelu Lukaku got four as they ran riot, with the first goal after 31 seconds, and Cook said Bournemouth are on a mission to end a wretched start to the year.
"The goals we conceded were very soft and that’s something we’ve got to put right. Especially against Manchester City.
"Everyone was very disappointed after the game but we can’t dwell on it too much. Once we got back to training our focus moved on to the game against City.
"The manager’s preparations have been no different this week. We conceded six goals so I am sure he has spoken to the lads individually and he has spoken to us as a group, too. Hopefully we can put those mistakes to bed and improve."
CONFIDENT CITY
Cook expects third-placed City to visit Dean Court full of confidence despite manager Pep Guardiola's view that their title challenge is over as they trail leaders Chelsea by 10 points.
"They had a dip in form like we’re having now which shows even the world-class teams struggle at times," said Cook.
"City play free-flowing football but it's up to us to play our own stuff and hopefully the fans here can get behind us and help us out of this run we’re in.
"How to stop City is a tough question because not many teams have been able to do it. But we’ve been working hard on the training pitch going through City’s strengths and weaknesses and we want to put that into action on Monday."
The Bournemouth defender is sure they can avoid a relegation fight with the likes of Sunderland and Crystal Palace.
"We’re confident that we won’t get dragged in," said Cook, who joined Bournemouth from Brighton in 2011 after several loan spells. “The aim at the start of the season was to stay up again and finish higher than we did last year.
"We're only three points off mid-table and one win would propel us back up there. It’s something we are looking to do very quickly because we need that win to change our fortunes.
"A mid-table finish would be great for us and we are not a million miles off that. There are 14 games left and a lot of those are winnable," he added.
(Editing by Ken Ferris)
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