A grassroots petition for a National/Green coalition - not authorised by either party - has gathered more than 3700 signatures.
Christchurch organic fashion entrepreneur Clive Antony, a National supporter, said he started the petition to initiate a "conversation" about a possible coalition of the two parties.
"It started to annoy me that everyone has been ruling out a National-Green coalition and rightly so as both parties have basically written it off," he wrote on the website change.org.
"But they have left the smallest of doors open. That's where we come in!
"I genuinely think there is common ground between the National Party and the Green Party, which could result in practical policy wins for New Zealand. Environmental issues such as carbon neutrality and social issues like child poverty come to mind."
Antony, who founded the organic fashion label Mallu as a social enterprise, said the idea came up in a conversation with a group of friends who between them voted for National, Green, Act and The Opportunities Party.
"We were thinking, why don't we just start a conversation?" he said.
"We shoved it on Facebook and it's going along really well, we are really stoked with the conversation that we are having about it. I know in my own personal circle it's been very positive and quite sane and civil."
Green Party leader James Shaw said on Sunday that he would not be being making contact with National, but he would take a call from National leader Bill English.
"It's my responsibility to do so. And we'll have to see what they've got to say," he said.
"But one of the things I will be saying in return is 'You know we campaigned on a change of government and you know what was in our manifesto ... and how incongruous that is to what the National Party policy programme is'."
English has not definitely ruled out an approach to the Green Party but said on Monday that he was expecting some signal from the Greens before that would happen.
"The Greens have consistently ruled themselves out of discussions with National," he said.
"We would need to see some indication of interest from them in constructive discussions and there hasn't been any indications so far."