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."