Senate Democratic leaders say they have reached an agreement on the party’s major climate and economic bill with Kyrsten Sinema – the centrist Democrat whose opposition remained a major hurdle to passing the most ambitious US climate legislation yet.

The support of Sinema, a former member of the Green Party who has evolved into one of Congress’ most conservative Democrats, was crucial to the passage of the bill, which tackles energy, environment, health and tax measures. Its success is seen as the Democratic party’s most substantive chance to deliver domestic policy progress before the midterm elections.

Backing from all 50 Democratic senators will be needed to pass any legislation in the evenly-divided Senate given the party’s narrow majority and Republican resistance to acting on the climate crisis.

Senate majority leader Chuck Schumer, said lawmakers had achieved a compromise “that I believe will receive the support” of all Democrats in the chamber. His party needs unanimity to move the measure through the 50-50 Senate, along with vice-president Kamala Harris’ tie-breaking vote.

Sinema, the Arizona senator seen as the pivotal vote, said in a statement that she had agreed to eleventh-hour changes in the measure’s tax and energy provisions and was ready to “move forward” on the bill.

She said Democrats had agreed to remove a provision raising taxes on “carried interest,” or profits that go to executives of private equity firms. That’s been a proposal she has long opposed, though it is a favorite of other Democrats, including the conservative West Virginia Democratic senator Joe Manchin, an architect of the overall bill.

The carried interest provision was estimated to produce $13bn for the government over the coming decade, a small portion of the measure’s $739bn in total revenue.

Securing Sinema’s support was the next challenge for Democrats after Manchin, the centrist Democrat famed for thwarting his own party’s climate goals, surprised Washington last week by backing the plan.

Manchin, who has made millions of dollars from his ownership of a coal-trading firm, made an abrupt U-turn last week and announced support for $369bn in spending to support renewable energy and reduce emissions.

Schumer has said he hopes the Senate can begin voting on the bill – known as the Inflation Reduction Act – on Saturday. Passage by the House, which Democrats control narrowly, could come next week.

Final congressional approval of the election-year measure would be a marquee achievement for Joe Biden and his party, notching an accomplishment they could tout to voters as November approaches.

Oliver Milman contributed reporting