Media are quoting ambulance workers as saying two of the people on the construction site in the west of the city were crushed by falling concrete.
It is not known how the third person died.
Police say seven people were taken to hospital and 150 homes were damaged.
"It seems there's been a couple of twister-like tornadoes that have come through and done the damage," a police spokeswoman said.
"We're still assessing the situation at the moment."
Eyewitnesses said they saw a tornado rip through the west Auckland suburb of Hobsonville, uprooting trees, blowing off roofs, and shattering windows.
Residents of Hobsonville spoke of seeing airborne trampolines, roofs and even a boat.
All flights out of Auckland airport have been delayed and the western motorway out of the city is closed.
Meteorologists say the storm system is moving to the south-east and more tornados are possible.
Construction worker Sam Nuttall said the storm, which officials said packed wind gusts of up to 110 kilometres per hour, struck without warning.
"You couldn't see it coming," he said.
"There was debris flying everywhere, steel flying around. Where I was we were sheltering between a big block wall and a truck and there were crates and everything flying at the truck, breaking the windows.
"I've never been in weather like that before."
A spokesman for the Metservice meteorological agency said it was too early to say whether the storm was a tornado.
"We've had no visual confirmation," he said. "The situation we have is a line of intense thunderstorms across Auckland and much of Northland."