The Greens have softened their demands over Labor’s planned $10 billion housing future fund, saying they would back the funding bill if it includes a rent freeze and at least $2.5 billion a year spent on new homes.

The revised offer, which halves the Greens’ original demand that Labor spend $5 billion a year on new housing, was almost immediately rejected by the government, which pointed out that placing a halt on rental increases was a state and territory matter.