WASHINGTON — One of the largest contributors to Senator Kyrsten Sinema’s political rise announced on Tuesday that it would cut off its financial support if the senator continues to refuse to change the Senate’s filibuster rules to allow for passage of far-reaching voting rights legislation.

Emily’s List, the largest funder of female Democratic candidates who support abortion rights, made the extraordinary announcement as the Senate barreled toward votes this week on a bill to reverse restrictions on voting passed by a number of Republican-led state legislatures.

If, as expected, Republicans block the bill with a filibuster, Democratic leaders plan to try to change the Senate’s rules to overcome the minority party’s opposition. To do that, Democratic leaders would need all 50 members of their caucus on board. But Ms. Sinema, Democrat of Arizona, has said she will not vote to change the rules, making her — along with another holdout from her party, Senator Joe Manchin III of West Virginia — a target of liberal activists’ ire.

“Understanding that access to the ballot box and confidence in election results are critical to our work and our country, we have joined with many others to impress upon Senator Sinema the importance of the pending voting rights legislation in the Senate,” Laphonza Butler, the president of Emily’s List, said in a statement. “So far those concerns have not been addressed.”

She added, “Right now, Senator Sinema’s decision to reject the voices of allies, partners and constituents who believe the importance of voting rights outweighs that of an arcane process means she will find herself standing alone in the next election.”

Emily’s List faced growing pressure from liberal activists and its own donors to take a stand ahead of this week’s showdown. The group was by far Ms. Sinema’s biggest donor in her run for the Senate in 2018, and potential primary challengers for her next run in 2024, such as Representative Ruben Gallego of Arizona, have begun making some noise.

Senator Elizabeth Warren, Democrat of Massachusetts, pointedly declined on Tuesday to rule out backing a primary challenge to Ms. Sinema.

“We’ll address that when we get past this week,” Ms. Warren said on “CBS Mornings” when pressed on the matter.

Senator Bernie Sanders, independent of Vermont, also hinted that he could support a primary challenge to Ms. Sinema or Mr. Manchin.

“Anybody who believes in American democracy has got to vote to enable us to go forward with 50 votes to suspend the filibuster,” he said on Tuesday.

Ms. Sinema said during a Senate floor speech last week that she supported the voting rights legislation, which would establish a national baseline for early voting, absentee ballots and ballot drop boxes; codify voter identification rules; promote donor transparency; and establish strict rules against the replacement of nonpartisan election officials without cause, among other provisions.

But Ms. Sinema said she would not support a single party unilaterally dismantling the filibuster, which requires the votes of 60 senators to cut off debate on most legislation. To do so, she said, would destroy any future for bipartisanship in the chamber.

Even if she yields to the pressure, Mr. Manchin could thwart the rules change anyway. And he was firm on Tuesday.

“You have a right to change your minds; I haven’t,” he said of his Democratic colleagues. “I hope they respect that, too. I’ve never changed my mind on the filibuster.”

Emily Cochrane contributed reporting.