Calls for Cook to step down from his role grew louder on Monday following England's 95-run defeat to India in the second Test at Lord's.
However, Moores is confident that the batsman is the right man to lead the Three Lions forward in the future.
"Cookie isn't hiding, he knows he's under pressure. He wants to carry on and is up for the challenge. I think he's the right man for the job," Moores told BBC Sport.
England return to action on Sunday when they get the third Test against India underway at The Ageas Bowl.