Labor and the Coalition on Monday finally struck a deal on a disclosure regime that will require politicians to publicly disclose their citizenship history and status by December 1, in a move designed to bring the citizenship crisis to a head.

In the Senate, Liberal Scott Ryan was elected President to replace Stephen Parry while One Nation's newest senator, Fraser Anning – who replaced Malcolm Roberts – sensationally quit the party after an hour in the job to become an independent.

As the eligibility crisis rolls on, with more MPs likely to be referred to the High Court and a final resolution, including possible byelections, likely to be months away, Coalition MPs are increasingly nervous the Turnbull government is terminal and an election defeat is inevitable.

Underscoring the damage that has been done to the government, a Newspoll published on Monday saw Mr Turnbull's standing as better prime minister crash and the Coalition fall further behind Labor in the two-party preferred vote to trail 45-55 – which, if repeated on election day, would see the government defeated in a landslide.

This is the 23rd consecutive poll in which the Coalition has trailed Labor.

Speaking in Manila on the sidelines of the East Asia Summit, a defiant Mr Turnbull claimed credit for the citizenship deal with Labor.

Asked what his message was to MPs who doubted his ability to revive the government's fortunes, Mr Turnbull said that"we face plenty of challenges in politics and we deal with them".

"Lets get on with it, get all the facts on the table and then the House can consider them and then, if they wish, refer people to the High Court. My job is keeping Australians safe, governing, delivering, ensuring our future prosperity, our economic security as well as our national security and when these big issues come up, resolving them, getting them sorted out," he said.

Senator Lambie is expecting to receive word from the British Home Office overnight on whether she is a UK citizen by descent through her Scottish father and has said she will quit immediately if advice indicates she is in breach of section 44 (i) of the constitution, which forbids dual nationals from being candidates for election.

Her Senate seat would then go to the next person on her Jacqui Lambie Network ticket – Steve Martin, the mayor of Devonport.

Constitutional law expert Anne Twomey told Fairfax Media that Mr Martin could himself fall foul of section 44 (iv) of the constitution, which forbids a person from being a candidate if they hold an office of profit under the crown.

Mr Martin told Fairfax Media he checked prior to the 2016 election that he was eligible to be a candidate but Professor Twomey said he could be disqualified as it could be argued the job of mayor was "under the crown".

The citizenship disclosure deal was done after Finance Minister Mathias Cormann struck a deal with Opposition Leader Bill Shorten that is modelled on the pecuniary interests register.

Politicians will be required to disclose their date and place of birth, citizenship at birth and, if they were foreign-born, details about when they naturalised as Australians.

They will have to include details about how they have satisfied themselves they are not dual citizens and what they have done to renounce any foreign ties.

They will also be required to disclose details about their parents and grandparents, given the risk of citizenship by descent.

While Mr Turnbull maintained on Monday that he was the right person to lead the government, Coalition MPs are despondent and few believe a recovery is possible – either under Mr Turnbull, or another leader.

As one senior MP bluntly put it on Monday: "people have up their minds about us".

Informal discussions have been held about whether to replace Mr Turnbull but, at the moment, there is no agreement on his replacement, with cabinet ministers Julie Bishop and Peter Dutton considered the most likely options.

Crucially for Mr Turnbull, there is little appetite for changing leader at this time.