The contenders are Deputy Prime Minister Bill English, Health Minister Jonathan Coleman and Corrections Minister Judith Collins. Several other National Party lawmakers said they might enter the race.
In New Zealand, the prime minister is chosen by the governing party's top lawmakers, who make up the caucus.
The caucus is expected to take a vote at a meeting on December 12. The new prime minister will have about 10 months to run the country before a general election next year.
Key has endorsed English, who is also the finance minister. English, 54, led the National Party 15 years ago for two years. He suffered a big defeat in the 2002 election, which was won by the rival Labour Party led by Helen Clark.
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Supporters say English has managed the economy exceptionally well under Key and would make for a steady hand as leader. Critics say he lacks the charisma needed for the top job.
English has also managed to return small surpluses on the government's books over the past couple of years.
English said he will deliver tax cuts and spread the country's wealth to where it's needed. He said he's gained wisdom and experience since last running the party.
"I was 39 years old then, with six children under 13," he said. "So if nothing else, I've got the opportunity to focus much more on the job now than was the case then."
"I was not a career politician so I've been a doctor," he said. "I live my life in the community. My kids go to state primary schools. So I think I have a deep connection with the concerns of ordinary New Zealanders."
He said there was an appetite for change within the party and he liked where he was positioned in the race.
Collins, 57, is known as tough, plain spoken and something of a maverick. In 2014, when she was Minister of Justice, she was forced to resign her portfolio three weeks before the general election due to her ties with a controversial blogger. She remained as a member of parliament.
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