Democrats needed at least 10 Republicans to overcome a legislative hurdle known as the filibuster that requires at least 60 votes in the 100-member Senate to advance legislation.
A vote to limit debate and thus end the filibuster was continuing on Wednesday, but with at least 41 "No" votes the Democrats' drive appeared dead to enact an election reform bill that they say is necessary to preserve the right to vote in the United States.
Several Republican-controlled state legislatures last year passed laws that Democrats fear will discourage voting, especially in areas with large blocs of Black, Hispanic and poor people who are part of the core of Democratic supporters.
Democratic senators may still vote on Wednesday to ease or suspend the filibuster rule in order to circumvent lock-step Republican opposition and pass the bill by a simple majority.
But conservative Democratic Senator Joe Manchin forcefully defended his opposition to changing the filibuster rule, even though he supports the voting rights legislation itself. His vote would be needed to overturn the rule.
"Let this change happen in this way and the Senate will be a body without rules," Manchin said.
"We don't have to change the rules to make our case to the American people for voting rights."
Biden told a news conference on Wednesday that he had not given up hope of advancing voting rights.
"We've not run out of options yet," Biden said.
With the Senate split 50-50, Democrats would need support from all of their caucus members plus a tie-breaking vote from Vice President Kamala Harris to change the chamber's rules on the filibuster.
Manchin's fellow conservative Democrat Kyrsten Sinema also opposes changing the rule.
