The government has pledged to bring on a vote for territory rights in the Senate by the end of the year.
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Independent senator David Pocock has been spearheading the legislation in the upper house, attempting to secure debate on the issue where possible.
The bill would remove the prohibition on the ACT and Northern Territory that stops them legislating on voluntary euthanasia.
Labor will not subject the bill to a guillotine motion - where they force a vote before every senator has had a chance to speak - due to it being a conscience vote.
"It's the government's clear position we resolve this bill once and for all by the end of the year," government leader in the Senate Penny Wong said.
"Restoring territory rights is an issue important to many in this place, particularly the territory senators."
Senator Pocock has been pushing for the vote to take place as quickly as possible, believing the bill has the numbers to pass the Senate.
The ACT senator has expressed concern that any delay in the legislation being brought to a vote could change the tide of support.
The bill, which passed the lower house 99 votes to 37 on August 3, will come back for debate in the Senate on Wednesday.
The Senate is scheduled to rise for the year on December 1.
Australian Associated Press
