Thames Valley Police said on Wednesday it had held discussions with the Crown Prosecution Service about allegations of misconduct in public office relating to documents recently released by the US Department of Justice.
Among the latest tranche of files released in the United States, emails suggested Mountbatten-Windsor, King Charles III's younger brother, shared official British trade documents with Epstein in 2010, after the financier's conviction for child sex crimes.
Andrew, now 65, was an official government trade envoy at the time.
"Allegations of misconduct in public office involve particular complexities, and therefore an assessment must be conducted carefully and thoroughly," Thames Valley Police Assistant Chief Constable Oliver Wright said in a statement.
"While we cannot provide timescales over when a decision as to whether a criminal investigation will be opened, we can assure you that Thames Valley Police is making progress as quickly as possible."
Earlier, Britain's chief prosecutor Stephen Parkinson told reporters the CPS was "in close contact" with police investigating Mountbatten-Windsor and also the United Kingdom's s former ambassador to Washington, Peter Mandelson.
Buckingham Palace said on Monday it was ready to support any police investigation into Mountbatten-Windsor after the release of the emails in the US.
Lord Mandelson is separately under investigation over alleged misconduct in public office, following claims he leaked market-sensitive information to Epstein while a government minister during the financial crash.
Epstein, who died by suicide in 2019 in a Manhattan jail cell while awaiting trial for sex trafficking, exploited his wealth and connections to cultivate relationships with prominent figures around the world over decades.
Andrew, the second son of the late Queen Elizabeth II, was forced to step down from a roving UK trade ambassador role in 2011, before quitting all royal duties in 2019.
He was then was stripped of his military links and royal patronages in 2022 by the King, amid allegations of sexual misconduct that he has always denied. He no longer lives on the Windsor estate.
