Scotland Yard has submitted a file to the Crown Prosecution Service relating to the prank call made to the hospital where the Duchess of Cambridge was treated for morning sickness.
Nurse Jacintha Saldanha, 46, was found hanging after she transferred the call from two Australian DJs to a colleague who then described Kate's condition in detail.
The Crown Prosecution Service (CPS) will now consider whether any potential offences may have been committed, Scotland Yard said.
Ms Saldanha, a mother-of-two, was found dead in her nurses' quarters at London's King Edward VII's Hospital by a colleague and a security guard on December 7.
Three days earlier she put through a call from DJs Mel Greig and Michael Christian, believing they were the Queen and Prince of Wales.
Ms Saldanha left two notes in her room and was found with marks on her wrist when her body was discovered, Westminster Coroner's Court in London heard as an inquest was opened and adjourned.
Greig and Christian spoke of their grief on Australian television soon after the nurse's death.
They said their prank had prompted "a tragic turn of events no-one could have predicted or expected".
A Scotland Yard spokesman said: "Following the death of Jacintha Saldanha, officers have liaised with the CPS as to whether any criminal offences had been committed in relation to the hoax call made to King Edward VII Hospital in the early hours of Tuesday December 4.
"On Wednesday December 19, officers submitted a file to the CPS for them to consider whether any potential offences may have been committed by making the hoax call."
On Monday, Ms Saldanha's children, Junal, 16, and Lisha, 14, were consoled by their father Benedict Barboza at a funeral service in Shirva, the small town in India where the nurse was buried.
Hundreds of mourners turned out in a show of support for the family who journeyed back to their home-town for the ceremony, around 30 miles north of Mangalore on the south-west coast.
Southern Cross Austereo (SCA), the parent company of 2Day FM, has ended the DJs' Hot 30 show and suspended prank calls across the company in the wake of Ms Saldanha's death.