John Wilson put his name to Labour welfare spokeswoman Jackie Baillie's proposal, allowing it to move to the next parliamentary stage.
Ms Baillie said: "While pressure from Labour has already forced the Scottish Government to offer some money to mitigate the effects of the bedroom tax, it is clear that this is nowhere near enough.
"The cost of mitigating the bedroom tax is estimated by Shelter to be in the region of £50 million each year.
"The SNP must find the resources within its £30 billion budget to fully mitigate the cost of the bedroom tax and protect council and housing association budgets.
"I will now be taking forward my final proposal for a bill and I hope that, with support from MSPs on all sides of the chamber, the Parliament will agree to legislate to ensure that no-one is forced to leave their home due to the bedroom tax.
"There is, of course, an easier way: the SNP could simply ensure the resources are on the table and there would be no need for the Bill to proceed."
The final proposal follows a three-month consultation which attracted 82% support for the introduction of a statutory no-eviction guarantee for tenants who fall into arrears solely due to the controversial UK Government policy.
The Bill would apply to all social housing tenants, covering local authorities and housing associations in Scotland.
The Westminster welfare reform, called a tax by opponents, cuts the amount of benefit people can get if they are deemed to have a spare room.
Mr Wilson, an SNP MSP for Central Scotland, said: "I and the SNP are happy to consider any constructive proposal to tackle Westminster's iniquitous bedroom tax
"I have supported Jackie Baillie's proposal in principle, in order that it can proceed as a Member's Bill, allowing the Parliament's non-executive bills unit to draw up more detailed proposals for consideration by MSPs.
"My position on the Bill will be determined by the detail, the evidence and the parliamentary scrutiny that is applied.
"Ultimately, the only way to deal with the Bedroom Tax is to scrap it completely, something that only the powers offered by a Yes vote in next year's referendum will enable us to do."