The legacy has drawn public praise still reeling from the dog's death.
EmJay Dette posted: "Best tribute to her brother Gazza."
"Great to here the blood line is still there RIP GAZZA," wrote Kelvin Riddick.
Police were yet to choose which pup would eventually join the region's dog team but said the public would be able to watch it grow up and progress through a series of photos on the Facebook page.
In the meantime a competition was being held to name the pup. In accordance with protocol its name must start with the letter K.
Gazza was shot last Friday morning by Pita Tekira who was on the run after ditching his electronic monitoring bracelet.
It is believed Tekira took his own life sometime between Friday night and Saturday morning, bringing Porirua's 26-hour siege to an end.
A police spokeswoman said planning for a memorial service for Gazza was also underway.