Labor and the crossbench could collaborate to force the establishment of a federal anti-corruption watchdog in the final parliamentary sitting weeks of 2018, unless the government agrees to a bipartisan deal.

Such a vote, if successful, would be the first major defeat for the Morrison government since the victory of independent Kerryn Phelps in the Wentworth by-election robbed the Coalition of its majority in the Lower House.

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Labor and the Greens have long supported a national version of the ICAC in New South Wales, which has exposed serious corruption in the state and led to the criminal prosecution of politicians.

Momentum for a national watchdog is growing. The independents on the six-member crossbench have developed their own 200-page draft legislation – yet to be publicly released – that would set up a wide-reaching body to regulate 21 government agencies.

It would have various investigative powers and would cover the murky area of political lobbying.

While attorney-general Christian Porter has previously dismissed the idea, saying there was no “persuasive evidence” that existing mechanisms were failing, he has recently said he was “open to discussing all policy options”.

In a letter to the prime minister on Monday, opposition leader Bill Shorten said it was time for the government to get involved in drafting a bipartisan proposal.

“Unfortunately, despite acknowledgement that the government is considering such a reform, your government has failed to engage with Labor on this issue,” Mr Shorten wrote.

“It is my strong view that bipartisanship is important if we are to get the reform right, and ensure the new body is above politics, as it should be.”

Mr Shorten went on to suggest Labor could move ahead with the crossbench in the final sitting fortnight of 2018, which begins November 26.

“I believe the Australian people would welcome our cooperation on this issue. However, while we are willing to work with you, we will not wait for you. We are willing to do what is necessary to pursue this reform in the coming weeks.”

Most of the Lower House crossbench strongly support the reform. Cathy McGowan is spearheading the crossbench bill, supported by the Green’s Adam Bandt and independents Andrew Wilkie, Rebekha Sharkie and Kerryn Phelps.

Queensland crossbencher Bob Katter is the only member of the crossbench who remains somewhat undecided, recently telling Fairfax Media he was “not sure”.

Mr Katter’s vote could end up being decisive in the hung parliament, where any push to defeat the Coalition in the chamber would need the support of every non-government MP.