"My position remains the same and I'm desperate to carry on because I love being England captain and it is an honour," Cook told reporters ahead of the third Test against India starting on Sunday in Southampton.
"Until the time where I don't feel I can carry on or someone taps me on the shoulder that's what I'll do."
Cook, beaten in seven of the last nine Tests in charge and going through a dreadful run of form with the bat, also shrugged off increased calls for him to go.
Kevin Pietersen, sacked by England in February, is the latest player to say Cook should quit because he isn't up to the job.
Cook added: "The last three or four weeks everyone has been saying that, so he's no different to anyone else.
"I'd much rather them say nice things about me but they've been through it themselves and they have been criticised in the past.
"I'm a strong-willed guy who doesn't take much notice of what is said from the commentary box or in the media.
"I've just got to be true to myself. It would be good for me to get through this tough period as a person, a captain and a leader."
Cook also said he had the support of the dressing room, saying: "Every guy who has spoken to me has given me their support, unless they're lying to my face."
The beleaguered batsman remains without a century in his last 27 innings but he is confident he can turn his form around.
"I've been putting in the hard yards without getting any rewards, which is incredibly frustrating," Cook said.
"The ball has been hitting the middle of the bat and I feel like I've been moving my feet well and it was just two mistakes that got me out (in the last test) at Lord's. To me it's not far away but I need to prove that by scoring runs.
"At the beginning when it was going really well and I was scoring runs people were saying being captain was helping my batting but now it's not going well people are saying it's not helping and that can be frustrating."
Stuart Broad will be fit to play at The Ageas Bowl but Chris Jordan could replace Ben Stokes in the only other change after Jos Buttler came in for the injured Matt Prior.
India lead the five Test series 1-0.
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