Prime Minister Tony Abbott has repeatedly refused to deny or confirm the reports despite growing pressure for answers from Indonesia, the United Nations, Labor and the Greens.
Labor leader Bill Shorten has told Fairfax Media the secrecy over the allegations is breathtaking and he has called on Mr Abbott to tell Australians what happened.
Labor immigration spokesman Richard Marles wrote to the Auditor-General on Sunday night requesting an investigation into whether the payments were made.
"If this happened, there are serious questions about the legal basis upon which it has happened," Mr Marles told ABC radio on Monday.
Greens senator Sarah Hanson-Young will ask the Australian Federal Police to investigate the claims and whether any laws have been broken.
She says Australians have a right to know whether taxpayer funds have been used.
"(Mr Abbott) doesn't have a mandate to break the law," Ms Hanson-Young told ABC radio.
"He doesn't have a mandate to hand out wads of cash on the ocean to people smugglers."
Meanwhile, Foreign Minister Julie Bishop has urged Indonesia to enforce sovereignty over its own borders, after Indonesia's Foreign Minister Retno Marsudi sought an explanation from Australia's ambassador Paul Grigson over the allegations.
Ms Bishop told The Australian the best way for Indonesia to resolve its concerns about Australia's operations to stop asylum seeker boats was to "enforce sovereignty over its borders".
Mr Marles says the claims are affecting Australia's relationship with Indonesia.
Asked whether he agreed with Ms Bishop that Indonesia needed to get its own house in order, Mr Marles said: "I don't think now is the time for the Australian government to be walking down that path."
The Australian Greens on Monday will try to win Senate support for a motion requesting the government table documents detailing any payments to individuals on board asylum seeker boats.
Passengers tell UNHCR the crew received a payment
The United Nations refugee agency says it's interviewed the 65 passengers from the boat who claim the crew were paid by Australian officials to return to Indonesian waters.
The UNHCR's James Lynch has told the BBC the passengers were transferred to a customs boat for four days before being put on two boats and sent back to Indonesia.
"The boat that was rescued by the Indonesian navy on 31 May - we have interviewed the 65 passengers and they have said that the crew received a payment," Mr Lynch said.
Indonesia's Foreign Minister Retno Marsudi sought an explanation from Australia's ambassador Paul Grigson on Saturday over the allegations.
Mr Abbott has repeatedly refused to confirm or deny the allegations, or whether the government would investigate them.
"The only question that matters is, is this government prepared to do what's necessary to keep the boats stopped?" he told reporters in Canberra on Sunday.
"The answer is yes."
He said it was important for Indonesia to know the government was "absolutely resolute" in its determination not to allow the people smuggling trade to resume.
Immigration Minister Peter Dutton refused to comment on the allegations on Sunday after denying them earlier in the week.
"From day one we have not commented on operations," he said, following the line taken by Mr Abbott and several other senior ministers on Operation Sovereign Borders.
Opposition Leader Bill Shorten said Australians deserved an answer following Mr Dutton's backflip.
"It is now time for Mr Abbott to make it clear ... has Australian taxpayer money been paid by the Abbott government to criminal people smugglers or not?" he told reporters in Canberra.
Opposition immigration spokesman Richard Marles said the claims were affecting Australia's relationship with Indonesia.
If the claims were true, Mr Dutton would have a "serious problem".
"If he has denied this up front and it turns out to be true, Peter Dutton is seriously injured," Mr Marles told ABC TV.
Finance Minister Mathias Cormann said the proposition that Mr Abbott's refusal to deny the allegations translated into payments having been made was "just wrong".
Former Howard government treasurer Peter Costello said it was appropriate that the government neither confirm or deny the reports.
"There's a lot of stuff that happens in the security area which a government neither confirms nor denies - that doesn't mean it happens," Mr Costello told Network Ten's Bolt Report.
"You don't want the smugglers to know what you're doing."
Greens Leader Richard Di Natale said any government payments to people smugglers would effectively put Australia in the people smuggling business.
