Climate Change Minister James Shaw said he would be disappointed if the Prime Minister grants new permits for gas and oil exploration.
Exploration permits are granted by the Government in an annual event known as the "block offer", which is usually held in April.
On Newshub Nation last week, Jacinda Ardern said the Government hadn't decided whether go ahead with this year's block offer: "That's something we're working on right now but we're not going to preempt that decision."
On RadioLIVE Drive on Monday, Mr Shaw said that while it was important for the Government to "take the time to get it right," he would prefer the block offer didn’t go ahead.
The Prime Minister on Tuesday delayed a meeting with the Indonesian Prime Minister to personally receive a 45,000 signature petition by Greenpeace calling for an end to oil drilling.
"It is very significant that the Prime Minister came and accepted the Greenpeace petition to end oil exploration, and it's a positive signal that she's actively considering the issue," Greenpeace climate campaigner Kate Simcock told Newshub Nation.
Environmental activist and actress Lucy Lawless said she thought Ms Ardern probably supported the petition, but was constrained by political obligations.
"I think it's about getting her colleagues entirely onboard. It's not that she doesn't want to, she's just not an autocrat."
Ms Lawless said you could be "forgiven for thinking" that the Green Party had gone soft on environmental issues now that they are in Government and that she had been disappointed in the Greens "for years now".
"Those of us who understand the knock-on effects of climate change wanted this 10 years ago."
Even if a moratorium on new exploration was put in place, existing licences would not be cancelled. Licences can be granted for around 20 years, and licence holder has the right to begun drilling anytime in that period.
The cost of putting a moratorium on new coal and oil mining has been estimated at $15 billion.
Cameron Matchwick from the Petroleum Exploration & Production Association New Zealand told RadioLIVE Drive that the petition from Greenpeace represents nothing new and that the 45,000 signatures "only represent 1 percent of the population".
Mr Matchwick says he accepts the Government has made a commitment to make the transition to a lower emission economy and that he thinks his industry "can play a strong role in that transition."
The oil and gas industry employs around 11,000 people each year in New Zealand.
He said that if the block offer doesn't go ahead, New Zealand's reserves of oil and gas would run out within a decade, and that gas, in particular, produces half the emissions of coal.
"Every New Zealander in one way or another relies on these products."