After meeting with vice president, Texas Democrats concede federal voting rights battle is in Senate

Schumer said one of the key voting reform bills will get a floor vote next week.

Following their "very robust conversation" with Vice President Kamala Harris, Democratic state legislators from Texas conceded that while the Biden administration wants to see federal voting reform legislation make it to the president's desk, the hurdle to passage is the U.S. Senate -- but they aren't giving up on conversations with those lawmakers.

"We know it's a priority for the administration but quite frankly it lies in the Senate," state Sen. Carol Alvarado, the chair of the Texas Senate Democratic Caucus, said after the meeting at the White House Wednesday.

Harris met with more than a dozen Democratic lawmakers from Texas, applauding them as "courageous leaders" for staging a walkout in the final hours of the state's regular legislative session last month. Their departure resulted in a lack of quorum, effectively killing a sweeping election law overhaul that would've imposed new restrictions on voting.

"We are not asking for the bestowal of a right. We are talking about the preservation of the right, that is the right of citizenship," Harris said. "When I look at that, and the -- the fact that Americans are at risk of losing their access to their right ... we know we have a great challenge in front of us and therefore fight."

With the state lawmakers seated around a table in the Roosevelt Room with her, Harris reiterated her and President Joe Biden's "absolute commitment" to seeing Congress pass two bills -- the John Lewis Voting Rights Act, or HR 4, and the For The People Act, or HR/SR 1.

Senate Majority Leader Chuck Schumer said Tuesday that he would bring the Senate's version of the "For The People Act" to the floor for a vote next week, but it's all but certain to fail without changing the Senate filibuster rules as no Republicans support it, and there is also one key Democratic holdout: moderate Sen. Joe Manchin of West Virginia.

Because of him, the bill brought to the Senate floor could look very different from the version passed by the House. Manchin told ABC News that last week, he delivered a list of items he supports and opposes in the current version of the bill. Schumer has said he will look at that and decide what changes he'll make to the bill.

Manchin previously told ABC News he supports the John Lewis Voting Rights Act, which is the more measured of the two bills and would restore key provisions in the original Voting Rights Act of 1965 that were struck down by the Supreme Court in 2013.

The Texas lawmakers met with members of the Democratic caucus on Capitol Hill Tuesday, including Speaker Nancy Pelosi and Schumer, but they did not meet with Manchin.

"There have been requests, there have been attempts and we are continuing to make those requests," said Alvarado, noting that two state lawmakers did meet with his chief of staff Tuesday.

Manchin told reporters on Wednesday he had planned to -- and was "hoping" to -- meet with the Texas delegation, but had a scheduling conflict. An attempt to schedule something Wednesday morning also did not work out.

"We're not going to give up on him because we believe that although he has other things in his district to take care of, we believe this is paramount to the people of the United States," state Rep. Senfronia Thompson, the dean of the Texas House Democratic Caucus, told reporters.

State Sen. Royce West said that "as legislators, we respect the legislative process," but urged political leaders, and the nation as a whole, to get behind their diverse coalition working to protect voters' access to the ballot.

"We stand together. You have Anglos, Hispanics, African Americans, all of us are standing together," West said. "We look like America, and we need other people to stand up."

ABC News' Trish Turner and Alisa Wiersema contributed to this report.