Anthony Albanese has been labelled "incoherent" and urged to refer himself to a corruption watchdog after launching an incendiary spray over flight upgrades.
The prime minister accused Opposition Leader Peter Dutton of being "obsessed" with attacking him and his family, in a heated back-and-forth between the party leaders.
But Mr Dutton returned fire on Tuesday, suggesting Mr Albanese was "not a good person under pressure" and he had repeatedly breached ministerial standards.
The barbs were traded over Mr Albanese's relationship with Alan Joyce, after reports emerged in a book that he had received 22 upgrades from economy and would liaise directly with the former Qantas boss for them.
The prime minister said all of the flights had been within the rules for federal MPs and he had declared them.
But Mr Dutton said he had clearly breached ministerial standards, which prevent him seeking and encouraging gifts in his personal capacity.
"He's not coherent, because his story keeps changing and Australians now are starting to question the integrity and the truthfulness of their prime minister," he said.
"(We're) seeing a PM who can't lie straight in bed … there's nothing he's said that's coherent, other than he picked the phone up to Alan Joyce to request an upgrade of air fares."
Earlier, Mr Albanese said coalition MPs had received more flight upgrades from Qantas.
"Peter Dutton seems to be obsessed with making attacks on myself, my family, and to try and engage in this obsession," Mr Albanese said.
"Peter Dutton seems to be determined to just be arrogant and nasty every day.
"That's up to him, but I'll continue to do my job."
He challenged the opposition leader to explain flights he received from mining billionaire Gina Rinehart.
Mr Dutton, who said the prime minister only knew about those flights because he had declared them, called for Mr Albanese to refer the matter to the National Anti-Corruption Commission.
Opposition transport spokeswoman Bridget McKenzie called for Mr Joyce to be called before a Senate inquiry.
"I don't know of another transport minister who had a hotline to the CEO of Qantas to request freebies for family holidays. That's what we're dealing with here," she told Nine's Today program.
Workplace Minister Murray Watt said the prime minister complied with all of the rules for parliamentarians.
"If anyone looks at Anthony Albanese's record, he hasn't been delivering favours to Qantas,'' he told AAP.
"He has consistently taken on Qantas over a range of issues."
Independent senator David Pocock said the revelations about travel arrangements highlighted the need for change for political entitlements.
"There are very legitimate questions to be answered. There is a terrible, terrible look," he told reporters in Canberra.
"But I think this has to be part of the broader conversation around just how shocking and basically useless our lobbying laws are in this country."
