Eight year-old Phoebe Hanson wrote to BBC Radio 4's PM saying she worried how the "bongs" would be replaced while Big Ben is being repaired next year.
PM's editor, Roger Sawyer, told Phoebe: "It would be quite a task."
A Facebook post by Phoebe's father, Jon Hanson, about the exchange has been liked and shared thousands of times.
The "Big Ben" bell, after which the Palace of Westminster's Elizabeth Tower is more commonly known, is expected to fall silent early next year for the first time since 2007 for a three-year renovation project.
The work will cost £29m and involve repairs to the clock's hands, mechanism and pendulum.
Replying to Phoebe's letter on 12 December, Sawyer said he was "very much taken with your idea", but there could be some operational obstacles because "the bongs are live".
He explained: "You'd have to rush in after school each day (and at the weekend), rush home for tea, homework, a bit of chillin', then a quick sleep.
"And then - here's the hard bit - you'd have to rush in again at midnight, because there are live bongs again before the midnight news.
"That's an awful lot of work for someone who is still quite young. I wouldn't like to do all that."
Speaking to BBC News, Mr Hanson said: "She was listening to Radio 4 in the car and Eddie Mair was saying that Big Ben would stop chiming next year, and she said, 'Oh no, we have to do something about it. I want to write a letter.'
"She was absolutely stunned to get a reply, and was so pleased she took it to school and showed everyone.
"She takes things very literally as she is on the autism spectrum, so I think she still wants to do the bongs.
"I'm very proud of her. I work in IT and we try to increase our internet presence, but then she has stepped in and done it in one fell swoop."
Writing on his Facebook page, Mr Hanson called Sawyer's response "priceless".
"She thinks I should drive her up to Broadcasting House and back twice a day," he told others who had commented on his page.
Sawyer said that although he was surprised to receive the letter, he was aware that the bongs at the end of the programme "seem to chime with the younger audience especially - no pun intended".
He said Phoebe's hand-written letter - accompanied by a covering letter - suggested she could hum the Westminster chimes before playing her own instrument for the bongs, and that she could record everything on her own microphone brought with her to the studio.
"What I say to people who ask what the job [of working on PM] entails is that it requires a lot of determination and tenacity, but also what is vital is imagination and free thinking," he said.
"She has a great future because she has that ability to problem-solve in a free thinking way. Those are the kinds of skills that the BBC needs."
A girl who offered to replace Big Ben's chimes on BBC radio has been let down gently after an editor warned her of the long hours that would be involved.