Officers from the Serious Crime Branch at Ladas Drive believe dissident republicans were behind the tiger kidnapping.
According to the PSNI, the bogus postman produced a gun and demanded money after the family let him into their home in the west of the city on Wednesday morning.
The robber held two of them hostage while another member of the family was forced to go to his place of work to collect money.
On his return he gave the cash to members of the gang at Fallswater Street around 9.30 am.
The alarm was raised after the money was handed over to the robbers and they made their getaway..
After their ordeal, one of the family members was taken to hospital for a check-up, but was later released.
Police have appealed to anyone who saw a man, carrying a holdall bag, get out of a silver car in Fallswater Street, off the Falls Road.
He handed the holdall to another man, who then walked in the direction of Iveagh Street.
Anyone who witnessed the incident is asked to telephone detectives at Ladas Drive on 0845 600 8000 or pass information confidentially to Crimestoppers on 0800 555 111.