Her new role comes as the Senate enters a crucial month in the Democratic drive to enact the most extensive elections overhaul in a generation.WASHINGTON — Vice President Kamala Harris did not come to her role with a list of demands. She wanted to be a generalist, in large part to learn the political rhythms of a president she was still getting to know. In the first few months of her tenure, some of her portfolio assignments were just that: assignments.But on the matter of protecting voting rights, an issue critically important to President Biden’s legacy, Mr. Harris took a rare step. In a meeting with the president over a month ago, she told him that she wanted to take the lead on the issue.Mr. Biden agreed, two people familiar with the discussions said, and his advisers decided to time the announcement of Ms. Harris’s new role to a speech he delivered on Tuesday in Tulsa, Okla. In his remarks, the president declared the efforts of Republican-led statehouses around the country to make it harder to vote as an “assault on our democracy, ” and said Ms. Harris could help lead the charge against them.He also gave a blunt assessment of the task: “It’s going to take a hell of a lot of work.”Back in Washington, the president’s announcement has not clearly illuminated a path forward for Ms. Harris, whose involvement in the issue stands to become her most politically delicate engagement yet. Her new role comes as the Senate enters a crucial month in the Democratic drive to enact the farthest-reaching elections overhaul in a generation, including a landmark expansion of voting rights that is faltering in the Senate.Her office has not yet announced its plans, aside from calls Ms. Harris held with civil rights activists, including Derrick Johnson, the president of the National Association for the Advancement of Colored People, and a few scheduled meetings with prominent voting rights groups. Her advisers say she will take a wide-ranging approach to the issue by giving speeches, convening stakeholders and using the vice-presidential bully pulpit to raise awareness of the importance of the vote.“The work of voting rights has implications for not just one year down the road or four years down the road but 50 years from now,” Symone Sanders, the vice president’s senior adviser and press secretary, said in an interview on Wednesday. “The president understands that and the vice president understands that, and that’s why we will implement a comprehensive strategy.”The voting rights bill faces a more urgent timeline. The vast majority of the party has agreed to make the bill the party’s top legislative priority, and Senator Chuck Schumer, Democrat of New York and the majority leader, vowed to put it up for a vote later this month so any changes could be put into effect before the 2022 elections.With just weeks to go, it remains far from clear if it can actually pass. Because Republicans have locked arms in opposition, the only path forward would require all 50 Democrats — plus Ms. Harris, who serves as the tiebreaking vote in an evenly divided Senate — to support not only the substance of the bill, but changing the filibuster rule requiring 60 votes to approve major legislation, allowing it to pass with a simple majority instead.A handful of Democratic senators have expressed unease about changing the filibuster, while Senator Joe Manchin III of West Virginia and Senator Kyrsten Sinema, Democrat of Arizona, have been more adamant in their opposition.Mr. Biden has already pledged to sign the bill, which the House passed with only Democratic votes this spring. Known as the For the People Act, the bill would overhaul the nation’s elections system by creating new national requirements for early and mail-in voting, rein in campaign donations and limit partisan gerrymandering. But with the bill all but stalled in the Senate, Mr. Biden has repeatedly expressed concern over its future in his discussions with Democrats.The announcement that Ms. Harris would be working to move the bill forward took many on Capitol Hill by surprise. Ms. Harris and Mr. Schumer spoke on Tuesday — and had plans to hold a follow-up conversation late Wednesday, a White House official said — but it did not appear Mr. Manchin or Ms. Sinema were given a heads up.In a statement, Mr. Schumer said he welcomed Ms. Harris’s help navigating into law an elections overhaul that was “essential to protecting the future of our democracy.”Proponents of the voting legislation took her involvement as a sign that their attempts to build pressure not just on lawmakers, but the White House, were being felt.Senator Chuck Schumer, the majority leader, said he welcomed Ms. Harris’s help navigating into law an elections overhaul that was “essential to protecting the future of our democracy.”Erin Scott for The New York Times“It’s an interesting move given the long odds of anything getting passed and signed into law,” said James P. Manley, who served as a senior aide to former Senator Harry Reid of Nevada, the Democratic leader before Mr. Schumer. “There’s not a lot of cards to play right now, so it shows me they are going to try to raise the public temperature of this thing.”Others pointed out that even though Mr. Biden has decades of experience moving legislation through the Senate, Ms. Harris, the first woman and woman of color to hold her role, comes to the issue with an equally valuable perspective as the country grapples with the ways American policies have marginalized and mistreated Black people.“I think that Vice President Harris herself personifies the need for voting rights to be extended,” the Rev. Al Sharpton, who attended the speech in Tulsa, said in an interview. “When she’s on the phone or walks into an office, we’re looking at the reason we need voting rights.”Michael Waldman, the president of the Brennan Center for Justice at the New York University School of Law, said that the decision to elevate Ms. Harris as the face of the administration’s work on the issue was a pivotal moment for the Biden White House given the number of voter suppression efforts that were moving forward — 389 bills in 48 states and counting, according to a tracker maintained the Brennan Center.“It has been decades since a Democratic White House has made voting rights and democracy reform a central goal,” Mr. Waldman said, but he added, “the clock is ticking.”Ms. Harris’s impact on the hand-to-hand politics of the Senate is expected to be limited, but she often drew attention to voting rights during her four years as a senator. During her last year in the Senate, Ms. Harris introduced legislation that would expand election security measures, require each state to have early in-person voting periods and allow for an expansion of mail-in absentee ballots.In 2020, Ms. Harris was also a co-sponsor of the John Lewis Voting Rights Act, which would restore a piece of the Voting Rights Act of 1965 that relied on a formula to identify states with a history of discrimination and require that those jurisdictions clear any changes to their voting processes with the federal government. The protections were eliminated by the Supreme Court in 2013.Still, Ms. Harris, who spent a chunk of her time in the Senate running for president, was not known for building especially close relationships with colleagues, and Mr. Manchin and Ms. Sinema are no exceptions.Several Democratic aides who work closely with the senators scoffed on Wednesday at the idea that Ms. Harris, known as a staunch liberal, would be the one to persuade either moderate lawmaker to change the filibuster rule. Nor is Ms. Harris a likely candidate to broker the kind of compromise on the substance of the bill needed to persuade Mr. Manchin, the only Democrat who has not sponsored it, to back it.Ms. Harris’s attempts in February to nudge Mr. Manchin to back the White House’s proposed $1.9 trillion coronavirus rescue package is illustrative.Mr. Manchin was piqued when Ms. Harris appeared, without warning, on a television affiliate in West Virginia to promote the package before he backed it. Though a Democratic aide familiar with the matter, who asked for anonymity to speak candidly, said the episode was now “water under the bridge” it prompted cleanup by top White House officials.Mr. Manchin and Ms. Sinema’s offices declined to comment about Ms. Harris’s new role.Senate Republicans, meanwhile, are doing their best to kill the bill and blunt any Democratic attempt to change the filibuster rule, which would leave their party powerless to stop the passage of sweeping liberal priorities well beyond voting rights.At an event in his home state on Wednesday, Senator Mitch McConnell, Republican of Kentucky and the majority leader, argued that Democrats were inflating the impact of new state voting laws in an attempt to justify an unwarranted and chaotic slew of top-down changes to the way states run elections.“What is going on is the Democrats are trying to convince the Senate that states are involved in trying to prevent people from voting in order to pass a total federal takeover in how we conduct elections,” he told reporters. He said “not a single member” of his party supported the bill.Aware of the daunting path ahead, allies of the White House said that shepherding the bill through Congress was only one piece of the effort. Ms. Harris could be useful in helping ratchet up pressure on private companies, working with civil rights organizations, and engaging local communities over the importance of registering to vote.“She understands the need to engage in what I’d like to call kind of an ‘all of the above approach,’” said Representative Steven Horsford, Democrat of Nevada. “We can’t take anything for granted when we’re talking about having people’s voice heard at the ballot box.” More