Thames Valley Police said 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 US, emails suggested Mountbatten-Windsor, King Charles' 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 on Wednesday, local time.
"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 Crown Prosecution Service was "in close contact" with the police investigating Mountbatten-Windsor and also the UK'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 United States.
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.
