The push to construct an affordable and social housing fund has stalled amid a three-way Senate stand-off.

The Albanese government failed on Wednesday to force a vote to establish its signature $10 billion Housing Australia Future Fund which would use interest earnings of up to $500 million a year to build 30,000 social and affordable homes.

The Greens are using their balance of power in the Senate to bring Labor back to the negotiating table, saying the fund does nothing to help renters in the short term.

"We are trying to get Labor to wake up and understand how serious the rental and housing crisis is," Greens leader Adam Bandt said.

The minor party wants $5 billion invested each year indexed with inflation, as well as a national rent freeze.

Housing Minister Julie Collins says the government agreed to a number of concessions but the party remained firm in its opposition.

The government will amend the legislation to index the $500 million cap to inflation - which independent senator David Pocock pushed for - but only from 2029/30.

She also pointed to a number of measures for renters in Tuesday's budget, including increasing rent assistance and billions in other social housing measures that leverage investment and boost supply.

The maximum rent assistance rate will rise by 15 per cent, or $31 a fortnight - the largest increase in more than 30 years.

"These commitments, together with the changes on indexation, mean the Albanese government has addressed every single concern raised jointly by the Senate crossbench," she said.

"Senators who say they support more social and affordable homes need to stop the delays and pass the bill this week."

The Greens and opposition teamed up in the Senate to stare down Labor's attempt to force a vote on the fund this week, meaning it's unlikely to be voted on in coming days.

The coalition opposes the fund on the grounds it will be inflationary and lead to higher interest rates.

Homelessness Australia chief executive Kate Colvin said it was urgent for the legislation to pass but greater commitment was also needed to deliver more homes.

Jocelyn Martin, from the Housing Industry Association, said social and affordable housing had suffered from "prolonged under-investment".

She said the government's proposed housing fund was an important step towards managing affordability.

"Greater investment in new housing servicing this part of the market will assist in reducing the number of households experiencing housing stress," Ms Martin said.

Treasurer Jim Chalmers said the government still had levers to pull while the fund was set up, including tax breaks for build-to-rent properties and broader cost-of-living relief measures.

"I know that that won't make the pressures in the rental market disappear but it will help people get through while we try to build more supply," Dr Chalmers said.