"We are not doing this for ourselves but for the greater good of the people ... Is there any one of the royal family who wants to be king or queen?" Harry, 32, who is fifth in line to the throne, told the Newsweek magazine
"I don't think so, but we will carry out our duties at the right time."
Queen Elizabeth, 91, has been on the throne since 1952 and is currently the world's oldest and longest reigning monarch. Harry said they wanted to carry on her work but her family would not be "trying to fill her boots".
"The monarchy is a force for good," he said. "We don't want to dilute the magic ... The British public and the whole world need institutions like it."
Harry, along with his elder brother William and William's wife Kate, has become a prominent mental health campaigner, citing his own anguish and emotional struggles following the death of his mother in a Paris car crash 20 years ago and disclosing how he sought counselling to help him cope.
In the interview, Harry said how the decision to make him walk behind her coffin as the funeral cortege slowly made its way through the packed streets of London in front of huge crowds of mourners when he was just 12 had had a lasting impact.
"My mother had just died, and I had to walk a long way behind her coffin, surrounded by thousands of people watching me while millions more did on television," he said.
The prince, who is dating US actress Meghan Markle, last year criticised the media for intruding into her private life and he told Newsweek that he tried to live an "ordinary life" despite the huge global attention he attracts.
He said even if he were king, he would do his own shopping.
"I sometimes still feel I am living in a goldfish bowl, but I now manage it better," he said.
"People would be amazed by the ordinary life William and I live. I do my own shopping. Sometimes, when I come away from the meat counter in my local supermarket, I worry someone will snap me with their phone.
"But I am determined to have a relatively normal life, and if I am lucky enough to have children, they can have one too."
(Reporting by Michael Holden; editing by Guy Faulconbridge)