The resignation sets the scene for a dramatic conclusion to the year in Canberra and also increases the likelihood the eligibility crisis – which started in July – will extend well into 2018.

Prime Minister Malcolm Turnbull swiftly used Mr Alexander's demise to apply pressure on his opponents, urging Opposition Leader Bill Shorten to force at least three Labor MPs under a citizenship cloud to resign or be referred to the High Court. If more MPs resign or are referred to the court, a "super-Saturday" series of byelections in February or March beckons.

The Coalition and Labor are still bickering over a disclosure scheme designed to bring the citizenship crisis to a head, but behind the scenes the two leaders' offices are talking and hopeful of compromise.

Mr Alexander quit on Saturday after Fairfax Media earlier revealed he was a dual British citizen. The backbencher had not renounced the citizenship he inherited by descent, and the UK Home Office found no evidence to support the former tennis champion's repeated assertions his father had shed all UK ties.

"I have always believed that I am Australian and solely Australian," Mr Alexander said. "Australia is tired of this absurd situation. I don't have any degrees, but I have a degree in common sense and it doesn't make any common sense."

Asked whether a general election was looming, Mr Turnbull on Saturday responded with a simple "no".

The government was already down one seat in the House of Representatives after the High Court disqualified former deputy prime minister Barnaby Joyce.

Two crossbenchers have guaranteed support for the government on confidence and supply but have not ruled out siding with Labor on other votes.

The Coalition now has 73 seats, or 74 including the Speaker. Labor has 69, but could muster 74 with the support of all five crossbenchers. By convention the Speaker only votes in the event of a tie and his or her vote is to maintain the status quo.

Crucially, Labor is still short of the absolute majority needed to suspend standing orders and put a no-confidence motion that could force the government to fall.

However it is possible that in the final two sitting weeks of the year it will use parliamentary tactics to combine with the crossbench, and win votes on individual pieces of legislation – such as a banking royal commission or reversing cuts to penalty rates.

If successful, this would be seen as a sign the government has lost control of the Parliament and could trigger an election.

Speaking at the APEC summit in Vietnam, Mr Turnbull said Mr Alexander had done the right thing and demanded three Labor MPs under a cloud – Justine Keay, Susan Lamb and Josh Wilson – follow.

"Labor should now seriously consider whether their members who admitted they were UK citizens at the time they nominated, not only were but knew they were, whether they should do the right thing, resign from Parliament now and the byelections could be held on the same day as the byelection for Bennelong," Mr Turnbull said.

But when asked whether that extended to crossbench MP Rebekha Sharkie, who Mr Turnbull last week warned could be a dual citizen and may need to front the High Court herself, the Prime Minister was less definitive. Ms Sharkie is one of the two crossbench MPs guaranteeing supply and confidence to Mr Turnbull.

"I will leave that to her. She is an independent. It is up to every member, every member and senator has to be satisfied that they are eligible to sit in the Parliament," the Prime Minister said.

Ms Sharkie and the three Labor MPs renounced their dual citizenships before the 2016 election but did not get official confirmation until after they had been nominated or even elected.

Legal advice commissioned by the Liberal Party has found Ms Keay, Ms Lamb and Ms Sharkie would all fall foul of section 44 of the constitution, which forbids dual nationals from being elected.

Mr Shorten said the Turnbull government was "falling apart right in front of our eyes", while suggesting more Coalition MPs could be dual citizens.

"Labor is prepared to work on an agreed process that brings this issue to an end – but we will not accept something that is not fair dinkum or sees taxpayers pay more money to fix Turnbull's problems," Mr Shorten said.

Writs for a byelection in Mr Alexander's seat of Bennelong are expected to be issued on Monday and the poll held on December 16, with Mr Alexander pledging to recontest.

But he only has about 10 days to renounce his citizenship before nominations close – a tight timeframe.

Bennelong sits on a margin of 10 per cent, which is normally considered safe. Liberal Party figures are confident the seat can be retained through Mr Alexander's strong personal support, and believe the culturally diverse constituency will have sympathy for the dual citizenship situation.

Labor held the seat from 2007 to 2010, when former ABC star Maxine McKew snatched it from former prime minister John Howard.

Labor's state selection committee will likely select the candidate. But given the tight time frame, and despite presiding over a reduction in Labor's vote against Mr Alexander in 2016, sources said it was likely Lyndal Howison will run rather than a "star candidate".

Manager of Opposition Business Tony Burke predicted a swing to Mr Alexander in the snap poll.

"No matter which way you look at it, the government as of today is a minority government with a hung Parliament and the government lurching from crisis to crisis," Mr Burke said.

ABC psephologist Antony Green said the government had the numbers to remain in power but it faced "potentially embarrassing motions".

The latest developments are not expected to derail plans to legislate same-sex marriage by the end of the year, if a "yes" vote succeeds in the postal survey.