Keam also said Korean media had reported the man was or had been an international student in New Zealand.
A New Zealand police spokesman confirmed late yesterday afternoon that Interpol was liaising with authorities overseas.
"This matter is being treated as a priority. However, we are unable at this time to go into specific details of steps being taken while we work to gather further information."
The Korean Herald reported the man is believed to have flown to New Zealand on Monday afternoon, two days after the deaths.
According to Yongin Dongbu Police investigations, the suspect was last spotted on CCTV footage entering the victims' apartment on October 21 about noon and leaving about five hours later, the Korean Herald reported.
Video showed the victims arrived home about 2pm, and their bodies were discovered by the 55-year-old mother's sister late on Wednesday evening.
The newspaper reported that the victims suffered multiple stab wounds to their upper chests.
Local police confirmed a kitchen knife thought to have been used was found next to the bodies.
Yongin Dongbu Police confirmed that the suspect's stepfather - the husband and father of the victims - was murdered as well.
The 55-year-old man was found dead on Thursday at 4.05pm, in the boot of a rented vehicle parked in a condominium carpark in Gangwon Province, the Korean Herald reported.
The suspect had rented the car, which was found with a GPS tracker, police said.
"The suspect was found to have left for New Zealand on Monday afternoon, two days after the alleged murders," Yongin Dongbu Police Station detective Shin Dong-hyoun told the Korean Herald.
Shin declined to give further details on when an extradition request to Interpol would be made.
The two countries have an extradition treaty that was signed in 2001.
A member of the Korea-New Zealand Cultural Association, Diane Lee, said she had read in Korean media that the man flew to New Zealand on a one-way ticket.
Visitors usually were required to have a return ticket, so she wondered if the man had New Zealand residency or citizenship.
Korean Air operates direct flights between Seoul and Auckland.
Keam said he was "very much shamed and worrying" about the incident but he was confident the man would be found soon.
About 35,000 Koreans call New Zealand home, he said.
