The Abbott government faces at least three inquiries into whether six asylum-seeker boat crew members were paid $US30,000 by Australian officials to return to Indonesia.
Labor has referred the issue to the auditor-general, the Greens have sought a federal police investigation and Indonesian authorities have launched their own probe into the payments.
International law experts say the payments are tantamount to the Australian government funding people-smuggling.
Labor used the start of a fortnight of parliament on Monday to ask the prime minister and other ministers to confirm whether the money was paid.
Opposition frontbencher Tony Burke, who unsuccessfully moved a censure motion against the prime minister, said a one-word answer would settle the matter.
Mr Abbott said the government did not comment on “operational details”.
“This government does not feel the need to broadcast our intentions and our tactics to our enemies,” he told parliament.
Immigration Minister Peter Dutton said the answer was simple: “We stopped the boats and they will stay stopped under this government.”
“All of the people in the command structure in Operation Sovereign Borders … will do whatever it takes, within the law, to meet our international obligations to stop the people smugglers.”
Mr Dutton and Foreign Minister Julie Bishop last week gave outright denials when asked about the issue by reporters.
However, both declined to restate their denials in parliament.
Attorney-General George Brandis told the Senate that “all ministers in our government tell the truth”.
International law expert Don Rothwell says any payment was tantamount to people smuggling, but he doubted Indonesia would take any action against Australia.
Mr Burke said Australian taxpayers had a right to know whether their money was being given to people smugglers.
Labor immigration spokesman Richard Marles has written to the auditor-general seeking an inquiry.
The Greens have asked the Australian Federal Police to investigate the claims and establish whether any laws have been broken.
“Tony Abbott is treating this like he is waving away a parking fine,” Greens leader Richard Di Natale said.
An AFP spokesman told AAP the referral was being evaluated.
Nicholas Cowdery, a former NSW director of public prosecutions, said the AFP could face a conflict given that the agency also plays a role in Operation Sovereign Borders.
Ms Bishop’s comment that it was an “intelligence” matter has sparked questions about whether Australian Secret Intelligence Service agents were involved.
ASIS chief Nick Warner confirmed in 2012 his agents had “contributed intelligence and expertise leading to many significant, and unheralded, successes … which have disrupted people smuggling syndicates and their operations”.