President Biden said on Tuesday that he had directed Vice President Kamala Harris to lead Democrats in a sweeping legislative effort to protect voting rights, an issue that is critical to his legacy but one that sees little hope of success in a divided Senate.

“To signify the importance of our efforts, today I’m asking Vice President Harris to help these efforts, and lead them, among her many other responsibilities,” Mr. Biden told a crowd gathered during a trip to Tulsa, Okla. “With her leadership and your support, we’re gonna overcome again, I promise you, but it’s gonna take a hell of a lot of work.”

Mr. Biden was in Tulsa to deliver remarks marking the 100th anniversary of a white mob destroying a vibrant Black business district, a violent episode that was one of the worst outbreaks of racist violence in American history but one that went largely ignored in history books. Mr. Biden told the crowd he saw the protection of voting rights as one of the most fundamental — and most endangered — pathways to ensuring racial equity.

The president described the more than 360 bills in 47 states trying to tighten voting rules as “a truly unprecedented assault on our democracy.” But his decision to have the vice president take it on added another problem to an already unwieldy policy portfolio.

Ms. Harris has been tasked with examining the root causes of migration from Honduras, El Salvador, and Guatemala. She will visit Mexico and Guatemala next week.

Her foreign policy portfolio comes in addition to a host of other assignments, including, but not limited to: selling the American Rescue Plan, championing Mr. Biden’s infrastructure package, representing women in the work force, highlighting the Black maternal mortality rate, assisting small businesses, assessing water policy, promoting racial equity, combating vaccine hesitancy, and fighting for police reform.

For her part, Ms. Harris embraced the assignment in a statement issued shortly after Mr. Biden’s announcement.

“In the days and weeks ahead, I will engage the American people, and I will work with voting rights organizations, community organizations, and the private sector to help strengthen and uplift efforts on voting rights nationwide,” Ms. Harris said. “And we will also work with members of Congress to help advance these bills.”

Mr. Biden has focused on issues related to voting rights for much of his career, but he faces especially wrenching decisions when it comes to the voting rights legislation he has asked Ms. Harris to help shepherd through Congress.

Known as the For the People Act, the bill is the professed No. 1 priority of Democrats this year. It would overhaul the nation’s election system, rein in campaign donations and limit partisan gerrymandering. But after passing the House, it hit a wall of Republican opposition in the Senate.

One option for Democrats would be to ram the bill through on a partisan vote by further rolling back one of the foundations of Senate tradition, the filibuster. But at least one Democrat, Senator Joe Manchin III of West Virginia, remains opposed to the idea, potentially scuttling it.

In Tulsa, Mr. Biden seemed to express open frustration at the odds facing the bill.

“I hear all the folks on TV saying ‘Why doesn’t Biden get this done?’ ” Mr. Biden said. “Well, because Biden has a majority of effectively four votes in the House and a tie in the Senate, with two members of the Senate who vote more with my Republican friends,” a likely swipe at Mr. Manchin and Senator Kyrsten Sinema of Arizona, another moderate Democrat.

The reaction to Ms. Harris’s tough new assignment, at least, was bipartisan.

“Biden carrying on the longstanding American tradition of passing off the terrible/ impossible tasks to your VP,” Alyssa Farah, who served as a spokeswoman for Mike Pence during the early part of his vice presidency, wrote on Twitter.