Public prosecutors must decide whether the case is strong enough to bring charges over a stunt that was condemned around the world and fuelled concerns about media ethics.
Indian-born Jacintha Saldanha, 46, was found hanging in her hospital lodgings in London, days after she answered the hoax call from an Australian radio station, an inquest heard.
She put the call through to a colleague who disclosed details of the Duchess of Cambridge's condition during treatment for an extreme form of morning sickness in the early stages of pregnancy.
"Officers submitted a file to the Crown Prosecution Service (CPS) for them to consider whether any potential offences may have been committed by making the hoax call," London's Metropolitan Police said in a statement.
A CPS spokesman confirmed it had received the file, but declined to comment on the timing or nature of possible charges.
"That is what we will be considering," he said.
British Prime Minister David Cameron has described the case as a "complete tragedy" and has said many lessons will have to be learned from the nurse's death.
Australia's media regulator has launched an investigation into the phone call. Southern Cross Austereo, parent company of radio station 2Day FM, has apologised for the stunt.
Britain's own media is already under pressure to agree a new system of self-regulation and avoid state intervention following a damning inquiry into reporting practices.
The presenters who made the call, Mel Greig and Michael Christian, have apologised for their actions.