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The story - which originally appeared in the Huffington Post - says the mayors will be left off the speakers list at the event in September.
It says the Labour leader and four shadow cabinet members will be given prime speaking slots in Brighton.
Mr Corbyn said the conference agenda had not been drawn up yet.
Asked about the story on a campaigning visit to Morecambe, the Labour leader said he had "no idea" why The Sun was reporting it.
He said: "The Sun seem to be trying to run the Labour Party. Well I have news for The Sun - they don't."
On Wednesday, Labour's shadow equalities minister Sarah Champion quit her role over an article she wrote for the Sun about grooming gangs.
The Rotherham MP told the newspaper a week ago that "Britain has a problem with British Pakistani men raping and exploiting white girls".
She apologised for her "extremely poor choice of words," after earlier suggesting the Sun had altered what she had written, something denied by the newspaper.
According to Huffington Post political editor Paul Waugh, Mr Corbyn and four shadow cabinet members - John McDonnell, Emily Thornberry, Diane Abbott and Keir Starmer - will be pushed as the face of the party and given prime speaking slots at the conference.
This means there will be less time for other leading figures, suggests Waugh, adding that some in the party fear the leadership want "revenge" on Mr Burnham and Mr Khan for being critical of Mr Corbyn in the past.