He is wanted in Korea in relation to the murder of his mother, stepfather and 14-year-old half brother on October 21, Korean media reported.
South Korean police said the bodies of the man's mother and son were found with stab wounds in an apartment, and the stepfather's body was found in the boot of a car.
The arrest is the first step to allowing authorities to look at an extradition.
Joon Yi, who is acting for Kim, told the Herald that his client will not oppose an extradition.
Discussions had started with the authorities to get the process under way, Yi said.
"It is all just a matter of paperwork now," Yi said.
"He is due to reappear on December 1, but if the extradition can be organised earlier, then he will be gone."
Prosecutor Emma Hopkin said in court that Kim met all the criteria for extradition.
The charges he faced in Korea were of the "utmost seriousness", she said, and Kim had also shown a willingness to travel.
"There must be some concern for public safety," Hopkin said.
Kim was arrested in Auckland on Sunday evening in a building while his wife was in a car.
