"Voting for this bill doesn’t mean there will be assisted dying in the ACT or NT. It will simply give territorians the same right to decide on it as all other Australians. People living in Canberra shouldn’t be denied a right that people living on the other side of the border in Queanbeyan enjoy."
Senator Leyonhjelm secured debate and a free vote on his bill as part of the deal he struck with Mr Turnbull in 2016 to help revive the Australian Building and Construction Commission.
However, the Prime Minister said last week that deal only extended to the Senate, not the House of Representatives.
Senator Leyonhjelm warned that reneging on their agreement may have dire consequences for future cooperation between the crossbench and the government.
Mr Barr said Mr Turnbull needed to stand up to his party and let the bill be debated.
"I urge all senators to support this bill, and call on Malcolm Turnbull to show some leadership and allow this matter to be decided by a free vote in the House of Representatives as quickly as possible," Mr Barr said.
The Greens are also set to hold a rally out the front of Parliament House on Tuesday calling on senators to support the bill.
Greens leader Senator Richard DiNatale has previously tried to remove the voting restrictions on the territories, and the party will back the Leyonhjelm bill.
ACT Greens crossbencher Caroline Le Couteur said this was "the most promising effort in decades of advocacy" to see Andrews Bill repealed.
"The pressure is on to ensure a majority of senators vote in support of this bill. It is time that the Australian government repealed this undemocratic law," Ms Le Couteur said.
University of Tasmania law professor Margaret Otlowski - who wrote the book Voluntary Euthanasia and the Common Law - told an ACT parliamentary hearing last week territories should not be "disadvantaged" in their abilities to make laws for their citizens.
While the federal parliament has the power under section 122 of the Australian constitution to remove some or all of the powers it grants to self-governing territories, Andrews Bill was the first time it exercised that ability.
"I see no reason, in the federation, to be differentiating between the rights of states and territories on this kind of matter," Professor Otlowski said.
"My background is not constitutional law, and obviously we currently do have a differentiation. There is the Australian Capital Territory (Self-Government) Act, and the Northern Territory has its own, so there is a different legal footing. But, as a matter of principle, I see no justification why the way one lives one’s end of life should have anything to do with the constitutional basis of the elements of a federation."
Professor Otlowski said it "did not sit well" not to give Australian jurisdictions equal powers.
"I think it is inappropriate to be using that Commonwealth authority over territories to reach in to areas that deal so intimately with people’s lives at the end of life," Professor Otlowski said.