Australia on Monday told Indonesia to better secure its borders in a stinging rebuke after Jakarta demanded answers to allegations Canberra paid to turn back a boat of asylum-seekers.

The response from Foreign Minister Julie Bishop came after the Indonesian foreign ministry on Saturday said if the claims were true, it would be "a new low for the way that the Australian government is handling this issue", while asking for an explanation.

Claims that the captain and five crew of a boat, carrying migrants from Bangladesh, Myanmar and Sri Lanka, were each paid US$5,000 by an Australian immigration official to turn back to Indonesia were made to Indonesian police on Rote island in the country's east last week.

Australian Prime Minister Tony Abbott has refused to deny the allegations, with the opposition Labor Party writing to the auditor-general requesting an urgent investigation.

The Greens party was also due Monday to ask the upper house Senate to demand the government table documents detailing any payments.

Bishop suggested in an interview with The Australian newspaper that Indonesia was to blame for failing to properly manage its borders.

"I look forward to hearing the full results of Indonesia's investigation of the people-smuggling crimes committed in Indonesia," she said.

This, she added, should include "any breaches of passport and visa laws, and establishing whether the captains and crews of these boats are part of people-smuggling syndicates or are paid by them".

"The best way for Indonesia to resolve any concerns it has about Operation Sovereign Borders is for Indonesia to enforce sovereignty over its borders," Bishop said, referring to Australia's anti-people smuggling policy.

"Operation Sovereign Borders is necessary because Indonesian boats with Indonesian crews are leaving Indonesia with the express intention of breaching our sovereignty, facilitated by illegal people-smuggling syndicates."

Abbott's conservative coalition implemented a tough immigration policy after coming to power in September 2013 that included military-led efforts to turn back such boats, which mostly come from Indonesia.

The hardline policy also involves sending asylum-seekers that arrive by boat to camps on the Pacific islands of Nauru and Papua New Guinea despite strong criticism from rights groups. They are banned from settling in Australia even if found to be genuine refugees.

Only one boat with asylum-seekers has reached the Australian mainland since December 2013. Before the policy was introduced, boats were arriving almost daily, with hundreds drowning en route.

Abbott on Sunday said the key message for Indonesia was that his government was "prepared to do what's necessary" to prevent unwanted boats arriving in Australian waters.

Labor immigration spokesman Richard Marles said the claims were affecting bilateral ties which were already strained after Indonesia executed two Australian drug-smugglers in April, with Canberra recalling its ambassador in protest.

Asked if he agreed with Bishop, he replied: "I don't think now is the time for the Australian government to be walking down that path."