The Albanese government will put its $10 billion Housing Affordability Future Fund to the Senate for a vote this week, with or without the support of the Greens, in a first step towards creating a trigger for an early, double dissolution election.

Despite concessions made by the government to get the Greens on board, the minor party, whose support is needed to pass the legislation, was, as of Monday, still opposed on the grounds it does not spend enough on housing or offer up-front help for renters.