The four flag designs were selected in August by a government appointed Flag Consideration Panel made up of 12 people.
All four designs were selected from 10,292 submissions, which were whittled down to 40 by the same panel.
New Zealand’s citizens will vote for their favourite of the four in a referendum in September this year, before another referendum is held in 2016 to decide whether the country should switch from its current flag to the new option.
However, the four designs were met with mixed reactions from the public when they were revealed last week.
And fifth design, named the Red Peak and created by Wellington resident Aaron Dustin, has received widespread public backing.
An online petition to make the Red Peak one of the four options had received 35,990 signatures by Thursday.
Key said there was “a well set out process” and that cabinet would not include any more options in the referendum.
“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 was quoted as saying by the BBC.
New Zealand’s current flag was adopted in 1902 and contains the Union Jack. Key believes that the flag is too similar to Australia’s and does not reflect New Zealand well enough.
None of four of the options, nor the Red Peak, feature the Union Jack anywhere in their design.