The Greens are calling for major amendments to the Albanese government's key election pledge, with independent senator David Pocock also holding out for a larger commitment.
Asked if he was confident the legislation would secure the support needed this year, Mr Burke said he wasn't involved in those conversations, as it wasn't his portfolio.
"We know it will make a difference, we know it will help," he told ABC radio on Wednesday.
"We're in this situation dealing with the crossbench the way we are because ... the opposition have taken the view that they'll just vote no.
"We continue to talk in good faith with the crossbench and we'd be very happy to talk to the opposition if they wanted to join conversations in good faith."
With ongoing negotiations, progression on the bill is not expected until the next sitting week which coincides with the handing down of the federal budget in May.
The government needs the Greens and two crossbenchers in the upper house to pass legislation.
Housing Minister Julie Collins has previously warned people at risk of homelessness needed the policy up and running sooner rather than later.
The fund would be established along with the independent Housing Australia body and a National Housing Supply and Affordability Council, which would provide independent advice.
