The local police force said they received a report of a man behaving in an intimidating manner in the village of Wolferton, eastern England.
They did not name Andrew Mountbatten-Windsor in their statement.
"Officers attended, and the man was arrested on suspicion of a public order offence and possession of an offensive weapon," the police statement said.
The suspect is being held for questioning at a nearby police station.
The Telegraph report, which cited an unnamed source, said a man wearing a balaclava approached Mountbatten-Windsor, King Charles' younger brother, and shouted at him.
Mountbatten-Windsor had been walking his dogs at the time, and left in his own car with a member of his security detail, the report said.
He moved to the King's private Sandringham Estate in Norfolk after he was evicted from his longtime home near Windsor Castle earlier this year.
Mountbatten-Windsor is provided with an annual stipend by the King.
This includes an allowance made for the cost of private security but it is up to the former duke how he uses this, the Press Association understands.
Mountbatten-Windsor was stripped of his titles last year by Charles and was arrested in February on suspicion of leaking government documents to the late US sex offender Jeffrey Epstein.
Mountbatten-Windsor has denied any wrongdoing in relation to Epstein and said he regrets their friendship.
with PA and AP
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