Dimbleby, 74, who has chaired the topical discussion show since 1987, said it “will be a wrench to leave”, but “I plan to remain as busy in the years ahead as I have been up to now”.
The long-running flagship show features questions to politicians and other panellists from members of the audience.
Dimbleby said: “After 32 years in the chair, I have decided to stand down from Radio 4’s Any Questions? at the end of June.
“It has been a great privilege to have been in this role for so long and in the great broadcasting institution which is the BBC. It will be a wrench to leave. But the time feels right.
“For more than four decades I have been fortunate to combine presenting, reporting and interviewing for radio and television as well as writing books. I plan to remain as busy in the years ahead as I have been up to now.”
Radio 4 said a new presenter for Any Questions?, which is broadcast from a different location each week, will be announced.
Dimbleby previously presented its sister programme Any Answers?
The broadcaster is the younger brother of David Dimbleby, who recently stepped down as the chair of Question Time, with Fiona Bruce later taking over.
He is the son of the late broadcaster Richard Dimbleby, who was dubbed “the voice of the BBC”, narrating the Queen’s coronation in 1953 and Winston Churchill’s funeral in 1965.
BBC director-general Tony Hall said: “Jonathan Dimbleby has been an absolutely outstanding presenter of Any Questions? Over more than 30 years he has commanded the respect of the audience and panellists alike.
“His sharp intellect and chairing skills have made Any Questions? essential listening. While he may be stepping down from this programme, I am delighted that we are in discussion with Jonathan about future projects with the BBC. I shall miss his travels round the country enormously.”
Dimbleby began his career as a TV and radio reporter for BBC Bristol in 1969.
He later anchored general elections for ITV.
He was praised for his report on the Ethiopian famine of 1973 and made more headlines with his interview with the Prince of Wales in which Charles admitted adultery with Camilla.
Radio 4 Controller Gwyneth Williams said: “Jonathan Dimbleby is one of Radio 4’s most distinguished and familiar presenters and I shall be sorry to see him leave the programme.
“Radio 4 has benefited over so many years from the depth of his political knowledge and his great skill as a broadcaster. I know our audience trusts and values his experienced judgment each week as the hugely respected host of Any Questions?
“Personally, I will miss his acumen and enthusiasm as he hurtles around the UK each week engaging audiences in challenging and nuanced exchanges with political leaders.”