Chief Minister Andrew Barr has also told his federal counterparts to vote in support of repealing the Andrews bill, which would clear the path for the Legislative Assembly to pass right-to-die laws before the 2024 ACT election.

Federal Labor backbenchers Alicia Payne and Luke Gosling will on Monday introduce a private members' bill to repeal the Howard-era laws which quashed the NT's world-first assisted dying regime and prevented the territories from legislating on it in the future.

After numerous attempts to overturn the ban in the past two decades, the Payne-Gosling bill is seen to represent the best chance of ending a ban which critics argue has turned ACT and NT residents into "second-class citizens".

While Labor will allow its members a conscience vote, the government has agreed to set aside time for the debate - as the now Prime Minister Anthony Albanese promised ahead of the election.

As advocacy groups on both sides continue to lobby behind the scenes and encourage supporters to bombard MPs with emails and letters, Labor's territory representatives have made a public plea to their fellow parliamentarians.

"In 2022, it is unacceptable that citizens of the territories are second-class citizens," Canberra MP Alicia Payne said.

"It's time all Australians enjoyed equal democratic rights, and I urge all my parliamentary colleagues to support this bill."

Fenner MP Andrew Leigh, who has moved motions and bills in the previous two terms to overturn the Andrews bill, said Canberrans deserved the right to have assisted dying laws debated in their local parliament.

"We shouldn't be silenced due to an outdated law that no longer reflects who we are as a country," Dr Leigh said.

Bean MP David Smith has strong personal reservations about assisted dying, but was adamant the ACT Legislative Assembly should be allowed to have the debate.

"However, it is also critical that well-funded palliative care is accessible for all citizens," he said.

Finance Minister Katy Gallagher said: "After fighting for equal democratic rights for our territory for almost twenty years now, it is incredibly exciting to see our government introduce a private members bill to put an end to the decades of discrimination".

Independent ACT senator David Pocock, who had prepared his own territory rights bill, confirmed he would support the Payne-Gosling legislation and would encourage his upper house colleagues to follow suit.

"The long running injustice that has seen people in the ACT and NT have fewer rights than people living in the states must end," Senator Pocock said.

The Canberra Times has for the past year been calling for the Andrews bill to be overturned as part of its Our Right to Decide campaign.

The Payne-Gosling bill will be introduced during private members' time on Monday morning, then briefly debated during government business in the afternoon. The remainder of MPs' speeches will be held in parliament's federation chamber, before it returns for a vote.

Sources are confident the bill will easily pass the House of Representatives, with a vote expected as soon as this week. The numbers are expected to be much tighter in the Senate.

A majority of Labor MPs are expected to back the bill, however ACM, publisher of this newspaper, understands Employment Minister and Leader of the House Tony Burke will vote no.

Mr Burke was involved with an anti-euthanasia group in the mid-1990s and, while not yet a member of parliament, lobbied MPs to support the original Andrews bill. It is understood his position has not changed.

Supporters in the Labor ranks used parliament's first sitting week to lobby colleagues on their own side and across the aisle.

One source said many MPs appeared to have little understanding about the territory rights debate, suggesting a large number of parliamentarians had yet to make up their mind.

As the vote nears, interest groups on both sides of the debate are stepping up their campaigns to sway MPs into their corner.

ACM last week revealed the Catholic Church was urging MPs to oppose the bill, while the Australian Christian Lobby was encouraging people to email senators to urge them to vote against a "dangerous and politically tone-deaf bill".

Pro-euthanasia advocacy group Go Gentle Australia has also encouraged supporters to write to politicians, resulting in more than 2000 emails being sent as of late last week.

Go Gentle spokesman Steve Offner said it was "extraordinary" that opponents were trying to use the same tactics which had failed "spectacularly" in all of the states, where assisted dying was now legal.

"To pose that an attempt to restore democratic rights to 700,000 people is a controversial act ... it is quite dismissive of those people [in the territories]," he said.

Mr Barr argued the ACT Legislative Assembly was now a "well-established" parliament and more than capable of debating its own assisted dying laws.

The chief minister noted the ACT had the benefit of being able to learn the lessons from the states when drafting its own legislation.

"It is time for the Andrews bill veto to be removed, and for these sorts of decisions to be able to made in the territories in the same way they've been made in the states," he said.

Mr Barr emphasised the government would not rush the process of creating ACT assisted dying laws if the federal ban was lifted, saying a period of public consultation and a "thorough" committee inquiry would be held after any bill was introduced.

But he wanted to have laws passed before Canberrans headed to the polls in October 2024.