Federal senators want to examine how Australian navy vessels breached Indonesian territorial waters during Operation Sovereign Borders.
The Australian Greens want the Senate's Legal Affairs committee to examine the matter but will need the support of Labor to get the inquiry.
Greens senator Sarah Hanson-Young says the government's obsession with secrecy is tarnishing the navy's reputation.
"These incursions are very serious and Tony Abbott's explanations to this point have been woefully inadequate," she said.
Customs and Defence are already working on a joint internal inquiry but it's unclear whether this will be made public.
Immigration Minister Scott Morrison, Operation Sovereign Borders commander Angus Campbell, Defence Force Chief David Hurley and Customs boss Michael Pezzullo, giving evidence at another Senate hearing in late January, justified government secrecy over the operations.
But the Greens want the new inquiry to look at whether towing or turning back asylum-seeker boats was under way when the breaches occurred.
They also want to know whether the operations and procedures of Operation Sovereign Borders breach international law.
Mr Abbott has confirmed that on a number of occasions since the stop-the-boats operation began, Australian vessels have accidentally entered Indonesian waters.
Meanwhile, Opposition immigration spokesman Richard Marles is concerned the Abbott government is trying to bypass a recent Senate decision which vetoed temporary protection visas for asylum seekers.
The government is reportedly using what are known as temporary humanitarian concern visas, which prevent asylum seekers from gaining permanent residency.
This is despite Labor and the Greens teaming up in the Senate last year to veto the government's controversial temporary protection visas.
The visa system, dusted off for reuse by Immigration Minister Scott Morrison had not been used for a long time, Mr Marles said.
"It was conceived for a very specific purpose and it is now being used specifically to, in effect, get around a decision Australia's Senate made," he told reporters.
"That is of deep concern to me."
Customs has released an 18-page graphic novel aimed at discouraging Afghan asylum seekers from making the dangerous boat journey to Australia.
It warns asylum seekers in the Afghan languages of Dari and Pashto that people will not be given entry into Australia if they go via boat without a visa.
It shows illustrations of asylum seekers being bitten by mosquitoes in tents on Nauru and Manus Island.
Senator Hanson-Young described the material as in "poor taste".
"It's fear-mongering propaganda that fails to even acknowledge that the vast majority of refugees are fleeing war, torture and terror," she told AAP.
