Joyce this week was told by the New Zealand Government that he had New Zealand citizenship by descent - MPs in Australia are barred from having dual citizenship and the rule has already seen two Green MPs leave Parliament.
Labour leader Jacinda Ardern, apparently responding to Bishop's comments, tweeted that she knew "absolutely nothing about the Barnaby Joyce case until it broke in the media yesterday".
"I value our relationship with the Australian Govt highly. I won't let disappointing & false claims stand in the way of that relationship," Ardern tweeted.
Ardern said this afternoon that it was "highly regrettable" Bishop has made "false claims" about Labour.
"I have been utterly transparent about this situation," she said.
"I stand by my statements this morning that I knew absolutely nothing about the Barnaby Joyce case until it broke in the media yesterday afternoon."
Ardern said she had no knowledge about the Parliamentary questions lodged by Hipkins and had made it clear those questions were not appropriate.
"I greatly value New Zealand's relationship with the Australian Government. I will not let false claims stand in the way of that relationship.
"I would happily take a call from Julie Bishop to clarify matters."
Ardern said she had contacted the Australian High Commission to "register my disappointment" and would be meeting with the High Commissioner in Wellington later today.
Speaking in Australia's Parliament, Joyce said the New Zealand officials were prompted to look into it by queries from the NZ Labour Party, although Dunne has since confirmed officials were first prompted by an Australian journalist rather than Hipkins.
Hipkins had put in two parliamentary written questions to Dunne about the legal status of an Australian born to a New Zealand parent on August 9, just after an Australian journalist asked about Joyce's situation.
Ardern denied Labour had been trying to help its Australian counterparts on a dirt digging exercise, saying Hipkins had not known questions asked of him by an "acquaintance" in the Australian Labor Party related to Joyce. Hipkins had not known the law so had put in formal written questions to Dunne.
She admitted Hipkins had an inkling of what the general issue might be, given two Green MPs had had to stand down because they were found to have dual citizenship.
She said it was now clear the inquiries were prompted by the Australian journalist rather than Hipkins.
"However, Chris himself has acknowledged that had he known what those questions would be used for, he wouldn't have asked them. And I've certainly relayed to him that it's my expectation that we focus on our own election rather than matters that are of no concern to us.
He didn't know who was involved but I've made it clear to him that regardless of those circumstances this is not an issue that we should have been involved in."
NZ First leader Winston Peters, a former Foreign Minister, said the Hipkins' "hit" on Joyce was the equivalent of the underarm delivery.
"It is distasteful to see the New Zealand Labour Party colluding with the Australian Labor Party on what was a political hit job targeting the Australian Deputy Prime Minister."
He also criticised Internal Affairs Minister Peter Dunne for corroborating the information that Joyce was a citizen.
"This could do serious damage to New Zealand's relationship with Australia and the rights of New Zealanders in Australia."
Peters said Ardern must have known about Hipkins' actions, and it had seriously damaged relations between Australia and New Zealand.
It would be Kiwis living in Australia who would ultimately pay the price of Labour's political "idiocy", Peters said.
"Look at the collateral damage - it's us. And if you have any relations in Australia who are second-class citizens, you can now find out why."
Peters said if any of his MPs behaved like Hipkins they would be demoted right down the list.
"You cannot have people behaving in that rogue way, or think it's a smart idea. The truth is, she [Ardern] must have known."
Foreign Minister Gerry Brownlee said Bishop's comments were understandable given Hipkin's actions - which he described as "quite extraordinary."
"It's extraordinary that a New Zealand member of Parliament has allowed himself to be used by a party in a different country with an intent to bring another party in that country down.
It's quite extraordinary. I just don't think we've seen anything like this before."
Brownlee said such events did not help in building strong relations with other countries - and would make it harder to deal with Australia on issues such as the recent decision to charge New Zealanders the same university fees as other international student - a topic that was still under discussion.
Asked if Bishop's own comments were appropriate when an election in New Zealand was so close, Brownlee said it was a reasonable reaction in the circumstances.
He said Hipkins had misused the Parliamentary question system and it was clearly with the intent of bringing down a party in another country. It was up to Ardern to decide how to deal with Hipkins.
"But it is a very strange thing to have done and it does indicate a degree of influence on that member from well outside the Parliament."
Asked if he believed Hipkins' claim not to know Barnaby Joyce was the subject of the query from an Australia Labor Party, Brownlee said it would be surprising to agree to act on something without knowing what it was about.
He also appeared to cast doubt on Internal Affairs Minister Peter Dunne's claim that it was a query by an Australian journalist that prompted the Department of Internal Affairs to look at Joyce's case rather than Hipkins, saying it was Hipkins' questions.
Asked if he was saying Dunne was wrong, Brownlee said "I think you'll find it was his question that prompted the need for the New Zealand High Commission in Australia to approach Mr Joyce and say 'this is the situation.'"
The Australian journalist has said he put in questions on the matter prior to Hipkins' questions.
Brownlee said he last spoke to Bishop last Friday but had not spoken to her about Hipkins.
Joyce has asked the High Court to look at the case, saying he had not been aware he was a New Zealand citizen - it was automatic because his father was New Zealander.
