Federal Government backbencher John Alexander has resigned from Parliament, meaning the Coalition has, at least temporarily, lost its majority in the Lower House.

But Prime Minister Malcolm Turnbull shrugged off the loss, ruling out a general election and insisting the Government remains stable.

In the biggest blow yet to the Coalition, Mr Alexander conceded he is "most likely" a dual-British citizen by descent, because his father was born in the UK, making him ineligible to sit in Parliament.

He will now renounce any links to Britain he may have and plans to contest his seat of Bennelong in a by-election, likely to be held before Christmas.

"I have always believed that I am Australian and solely Australian," Mr Alexander said at a press conference on Saturday.

"In view of recent events and the High Court decision, in my position, I have had to thoroughly examine my situation.

"Given what I have learned about the Constitution and understanding now of the High Court decision just a couple of weeks ago, I can no longer, with sufficient certainty, maintain the belief that I have held through my 66 years."

Mr Alexander said he spoke to Mr Turnbull late on Friday and told him he intended to stand down.

"He encouraged me to be decisive, to act expeditiously, and to put things beyond any question of doubt and that's what I think the people of Bennelong want," he said.

"Australia is tired of this absurd situation."

Turnbull supports Alexander's decision

Speaking on the sidelines of the APEC summit in Vietnam, Mr Turnbull said Mr Alexander had "done the right and honourable thing" and tried to turn the spotlight on the Labor MPs who also have questions about their citizenship status.

John Alexander said Prime Minister Malcolm Turnbull encouraged him to act swiftly. ( ABC: Nour Haydar )

"Bill Shorten and Labor should now seriously consider whether their members who admitted they were UK citizens at the time they nominated should do the right thing, resign from parliament now and the by-elections could be held on the same day as the by-election for Bennelong," he said.

The twin resignations of Barnaby Joyce and Mr Alexander will test the Coalition's control over the House of Representatives for the final sitting fortnight of the year.

The Government was elected with a one-seat majority, but with the loss of the two MPs, it will now be forced to rely on crossbenchers Rebecca Sharkie from the Nick Xenophon Team, and Independent Cathy McGowan, for confidence and supply.

Labor has already declared it will exploit the Government's weak position in the Lower House, and possibly move motions to amend penalty rate laws or set up a banking royal commission.

In what is shaping up to be a potentially explosive final sitting fortnight, Manager of Opposition Business Tony Burke has put the Government on notice, warning Labor will be pursuing its own agenda.

"Our commitment is to represent the people who've been hurt through the banks without getting a royal commission and representing the people who've had their penalty rates cut," Mr Burke said.

While Mr Joyce has a strong chance of winning back his seat of New England, the Bennelong by-election will present a tougher challenge for the Coalition.

Mr Alexander holds Bennelong by a safe margin, and it has been a long-term Liberal seat, but Labor's Maxine McKew defeated then-Prime Minister John Howard in 2007.

He has conceded there is a chance he could lose, but said his time as a tennis player has taught him that "all you can concern yourself with is to play the game as best you can, and within the rules".