"I hope we can get this done, but I'm not sure," Mr Biden said after the meeting.
"As long as I have a breath in me, as long as I'm in the White House, as long as I'm engaged at all, I'm going to be fighting to change the way these legislatures have been moving."
After a trip to Atlanta, Georgia, Mr Biden spent Thursday trying to rally Senate Democrats at the US Capitol to unite and alter the upper chamber's rules to pass the bills.
But minutes before Mr Biden made the trip to Capitol Hill, key Democratic holdout Kyrsten Sinema plunged a dagger into the administration's hopes by voicing her opposition to change the "filibuster" rule that would allow Democrats to vote on and pass the voting rights legislation.
The Arizona senator said that the answer to divisiveness cannot be done by changing rules so one party could pass controversial bills.
“We must address the disease itself, the disease of division, to protect our democracy," she said, speaking from the Senate floor.
The Democratic bill intended to protect voting rights passed the US House of Representatives earlier on Thursday.
The Freedom to Vote Act and the John Lewis Voting Rights Advancement Act together would make Election Day a holiday, expand access to mail-in voting and strengthen US Justice Department oversight of local election jurisdictions with a history of discrimination.
Democrats see the voting rights bills as a last chance to counter new voting restrictions in Republican-controlled states before the November 8 congressional elections, when they run the risk of losing their narrow majorities in at least one chamber.
Holding only 50 seats in the Senate, Democrats remain divided on how to edge around the rule that has hampered them.
Senate Majority Leader Chuck Schumer on Wednesday outlined a strategy to ensure a floor debate on voting rights after three separate attempts last year were stymied by Republicans.
Under the plan, the House would repackage two elections-related bills into one and pass it. It would then go to the Senate under a special procedure preventing Republicans from blocking debate.
If Republicans remain opposed, that bill would not pass the Senate unless all Democrats agree to change the filibuster, he said.
Mr Schumer set Martin Luther King Jr Day as the deadline to pass the voting rights legislation and change the rules of the filibuster.
At least two Senate Democrats - Joe Manchin of West Virginia and Ms Sinema - are opposed to changing the rules.
Former presidents Bill Clinton and Barack Obama, as well as talk show personality Oprah Winfrey, have all reportedly phoned Mr Manchin convince him to go along with the plan, but Mr Manchin and Ms Sinema have both seemingly been impervious to any pressure thus far.
Senate Minority Leader Mitch McConnell has said Republicans remain opposed to the voting rights legislation and filibuster rule changes.
Mr McConnell also criticised Mr Biden for a speech in Atlanta on Tuesday pushing for an overhaul of the filibuster to pass voting rights bills, calling it "incoherent, incorrect and beneath his office".
In his Tuesday speech, Mr Biden said he is "tired of being quiet".
“The threat to our democracy is so grave, we must find a way to pass these voting rights bills, debate them, vote," he said.
Republican-led legislatures in 19 states have passed dozens of laws that critics say make it harder to vote.
Agencies contributed to this report