Labor’s plan to raise the tax rate on the largest super balances will become law after the Greens agreed to wave it through the Senate, ending a three-year political fight over concessions for wealthier retirees.

The Greens’ treasury spokesperson, Nick McKim, said the decision to support the watered-down package was a green light for Labor to pursue “bold reform” in the upcoming federal budget, including scaling back the capital gains tax discount and negative gearing.

The Greens confirmed its position before a scheduled debate in the upper house on Tuesday, guaranteeing Labor the numbers to legislate the changes this week.

“There is a massive Labor majority in the House of Representatives, the opposition is a rabble and the numbers are there in the Senate as long as Labor shows courage. The only limit is Labor’s level of ambition,” McKim said.

Under the superannuation tax changes, the concessional tax rate on earnings for balances between $3m and $10m will double from 15% to 30%.

Balances above $10m will be subject to a new, higher 40% rate.

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The expected passage of the laws will come three years after the prime minister, Anthony Albanese, and the treasurer, Jim Chalmers, first announced plans for “modest” changes targeting the largest superannuation balances.

After failing to pass the legislation in its first term, the government faced a fresh wave of criticism over the proposal to tax unrealised capital gains.

In October, Chalmers announced that contentious element would be axed and the $3m and $10m thresholds would be indexed to inflation, addressing a separate concern from critics.

The government also agreed to increase the low-income superannuation tax offset by $310 to $810 and raise the eligibility threshold from $37,000 to $45,000 from 1 July 2027.

At the time, the Greens welcomed the increase to the tax offset but lashed the other changes as a “capitulation to the wealthiest people in the country”.

The Coalition’s opposition meant the Greens were Labor’s only hope for passing the laws in the Senate.

But the minor party has decided against playing hardball, agreeing to support the revised package without demanding concessions in exchange.

McKim said the stance was a “down payment” on the Greens’ support for Chalmers and Albanese pursuing “genuine, progressive” tax reform in the 12 May budget.

Labor is considering winding back the 50% capital gains tax discount and potentially capping the number of properties that can be negatively geared as it weighs options to address international inequality in the housing market.

“This budget is a once-in-a-generation opportunity for ambitious tax reform, and we’re opening the door for Labor to walk through,” McKim said.

“The current tax system has turbo-charged the housing crisis, wealth inequality and a deepening intergenerational divide. We are demanding bold reform that responds to these challenges and makes our society fairer for young people and working people.

“Labor is standing in front of an open goal – if they bring reform that would rein in spiralling inequality and the housing crisis, they will have the Greens’ support to get it done. The only obstacle to real change is Labor.”