More than 20,000 people have signed a petition for the "Red Peak" flag design to be added to the four selected as possible alternatives.
The petition, on change.org, was launched after the Flag Consideration Panel last week unveiled the final four designs it had chosen from a longlist of 40.
But New Zealand Prime Minister John Key this morning ruled out adding a fifth flag.
"To accept any other flag we would have to change the law and we're not going back to parliament to change the law," he said.
Should 'Red Peak' be added to the final selection? Vote below
Kiwis are set to vote on their preferred flag design in a public referendum in November. A second referendum - to chose between the preferred alternate and the current flag - will be held in March 2016.
The Flag Consideration Panel has faced significant backlash since it revealed the final four, with many saying Red Peak, by designer Aaron Dustin, should be on the list.
The design features a red triangle in its centre with two white strips on either side. The left and right corners of the flag are black and blue, respectively.
Over the weekend, the petition for John Key to add Red Peak to the list amassed more than 20,000 signatures and large numbers of comments.
“I love this flag,” posted one supporter. “It makes me feel patriotic. It represents the New Zealand I want to live in, one with a deep history and a bright future.”
The hashtag #RedPeak was trending in New Zealand overnight.
In a statement on Sunday, a spokeswoman for the Flag Consideration Panel said there was nothing it could do.
"The panel was appointed by government to select the alternative which it has done and cabinet has approved those alternatives so they will go forward," she said.
"It's out of the panel's mandate and is now part of an official government process."
The Red Peak design is based around the Maori myth of Ranginui and Papatuanuku, the sky father and earth mother, who were locked together in an embrace before being prised apart by their children.