Independent senator Jacqui Lambie and Senator Nick Xenophon have also backed the motion.
The motion asks the government to "direct the Australian Bureau of Statistics to issue a statement declaring that no Australians will be fined for failing to complete the census".
The census was supposed to be held mostly online on 9 August but the website was taken offline at 7.30pm following what was claimed to be a distributed denial of service attack. This claim has been disputed.
Greens Senator Scott Ludlam, who had said on the day of the census that the party would, if needed, push for legislative amendments to ensure that fines were not imposed, said: "We've asked the government to assure Australians caught up in the censusfail that they will not be fined.
"It's ridiculous to think that thousands of people who have been unable to complete the census thanks to technical failures, or unwilling to after their confidence in the process was shattered, might be compelled to pay this government $180/day thanks to the government's own incompetence.
"When the senate reconvenes on 12 September, we'll vote on our motion, and if sense prevails, the Senate will demand that the government make those assurances."
Both the ABS and IBM, which received a $9.6 million contract to run the census, were blamed for the debacle.