With the Liberal-Nationals coalition firmly opposing the Housing Australia Future Fund, the government needs the votes of the Greens and two crossbenchers in the Senate to pass the bill.
In a letter to crossbench senators, Housing Minister Julie Collins said the government would make a series of changes to the bill, including dumping a $500 million cap for the amount of earnings distributed to housing.
Under the changes, there will be a fixed disbursement of $500 million a year from 2024/25, which will also be indexed from 2029/30.
The Greens decided at a meeting on Tuesday to reject the government's offer, but the party will continue negotiating.
The minor party is calling for more funding for social and affordable housing than promised under the current proposal, and for freezing or capping rent increases.
Greens leader Adam Bandt said his party had "bent over backwards" to compromise.
"Labor is refusing to move on two key issues that we've raised from the start ... making sure renters who are in crisis get looked after with a rent freeze and rent controls and spending some serious money on housing at a time when the waiting lists are growing," he told reporters in Canberra on Tuesday.
"They're not compromising - they're just clarifying - and the best that they can do is close a loophole in their bill that might have seen no money spent at all."
With the legislation failing to clinch crucial support, Senate business was reordered on Tuesday so the bill to set up the voice referendum is dealt with first.
Prime Minister Anthony Albanese told a Labor caucus meeting the Greens' rent cap proposal would effectively nationalise the private housing market and was "absurd and untenable".
Housing Minister Julie Collins told the meeting the federal government did not have the constitutional power to deal with rents, any policy change on rents would put downward pressure on supply.
Ms Collins said the government had moved on eight issues raised by the crossbench.
She underlined the fact $8 billion would be spent on housing and homelessness this financial year, on top of what the states and territories were spending.
The $10 billion housing fund was the single biggest investment in social and affordable housing from a federal government in more than a decade, Ms Collins said.
The fund would be invested to generate returns with the aim of subsidising 30,000 affordable homes a year.
The policy has the backing of key groups in the sector including the Property Council, Master Builders and the Housing Industry Association.
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