The New IRA is one of a small number of active militant groups opposed to a 1998 peace deal that largely ended sectarian violence in the region.
It has been behind many of the attacks on police, including a similar attempted car bombing at a police station outside Belfast last month.
The targeting of police officers at their homes would be an escalation of those attacks.
The last officer to be killed in Northern Ireland, Constable Ronan Kerr, died when a bomb exploded under his car outside his home 15 years ago.
The dissident group rejects the political compromises at the heart of the Good Friday Agreement that Northern Ireland will remain part of the United Kingdom unless a majority of the region votes by referendum to unite with Ireland.
Detectives investigating the attack on Dunmurry police station at the weekend arrested a 66-year-old man under terrorism laws, the Police Service of Northern Ireland said.
The New IRA claimed responsibility for hijacking a delivery vehicle and forcing the driver to take it to the police station on Saturday, the Irish News quoted the nationalist militant group as saying on Tuesday.
The organisation said in a statement, which the Irish News said it received and included a codeword to confirm its veracity, it had intended to kill police coming out of the station after the driver was told to shout that there was a bomb in the car as they left.
"It is our intention, if they keep harassing the republican people, to bomb them (police officers) in their own houses, with no warning," the statement added.
The New IRA, a far smaller organisation than the Irish Republican Army, which disarmed after the peace accord, typically claims responsibility for any attacks in coded statements to local newspapers.
Senior police officials announced a high-visibility policing operation across Northern Ireland in response to what they described as an ongoing dissident threat.
Assistant Chief Constable Davy Beck said the latest attack had demonstrated a clear intent to disrupt communities and potentially injure or kill police officers and staff.
