The Thames Valley Police said it was "assessing" reports that Andrew Mountbatten-Windsor sent trade reports to convicted sex offender Jeffrey Epstein in 2010.
The assessment followed the release of millions of pages of documents connected to a US investigation of Epstein.
The police force on Thursday did not name Mountbatten-Windsor, as is usual under UK law.
But when asked if he had been arrested, the force pointed to a statement saying they had arrested a man in his 60s.
Mountbatten-Windsor, whose birthday is on Thursday, is 66.
"Following a thorough assessment, we have now opened an investigation into this allegation of misconduct in public office,'' the statement said.
"It is important that we protect the integrity and objectivity of our investigation as we work with our partners to investigate this alleged offence.
"We understand the significant public interest in this case, and we will provide updates at the appropriate time."
Pictures circulated online on Thursday appearing to show unmarked police cars at Wood Farm on the Sandringham Estate in Norfolk, with plain-clothes officers appearing to gather outside the home of Mountbatten-Windsor.
There was no immediate comment from Buckingham Palace.
Andrew is the first senior British royal in modern history to be arrested.
The former prince, the brother of King Charles and the second son of the late Queen Elizabeth, has always denied any wrongdoing in relation to Epstein, and said he regretted their friendship.
But he has not responded to requests for comment since the latest release of documents.
Mountbatten-Windsor had been reported to police by the anti-monarchy campaign group Republic following the release of more than three million pages of documents relating to Epstein, who was convicted of soliciting prostitution from a minor in 2008.
Those files suggested Mountbatten-Windsor had in 2010 forwarded to Epstein reports about Vietnam, Singapore and other places he had visited on official trips.
with Reuters and PA
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