AUSTIN, Texas—At least 59 Texas House Democrats are fleeing the state of Texas Monday to deny the Republican-controlled state Legislature the quorum necessary to pass voting legislation during a special session, a source familiar with their plans said.

The Democrats were set to head to Washington, D.C., on two chartered airplanes, the person said. The legislators will have to remain out of state for weeks to avoid voting during the 30-day legislative session, which Republican Gov. Greg Abbott called after a previous walkout by Democrats shortly before a voting deadline killed the legislation at the end of the body’s regular session in May.

The bill that was considered in May would have set some of the nation’s broadest restrictions on voting. Mr. Abbott called the legislation a matter of election security, and an emergency. Democrats say the legislation is aimed at preventing their supporters, including many minority group members, from voting.

After Democrats killed the vote by staging a walkout this spring, Mr. Abbott called the special session for a legislature that typically meets for only five months every other year.

The Texas House requires at least two-thirds of its 150 members to be present in order to conduct official business. Any absentee lawmakers without an approved leave of absence may be arrested by the sergeant-at-arms or other law enforcement officers and forced to return to the chamber, according to state rules. Mr. Abbott has the ability to continue to call special legislative sessions, which are limited to the topic or topics that he chooses.

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“I do not know when we will return,” said Rep. Sheryl Cole, an Austin Democrat, in a text message. “It came to this because of Republican voter suppression.”

Rep. Julie Johnson, a Dallas-area Democrat, tweeted a photo of House members together on a bus. “When it comes to protecting our right to vote, all cards are on the table,” she wrote.

The governor called the Democrats’ move an abandonment of responsibilities in a statement Monday. “Texas Democrats’ decision to break a quorum of the Texas Legislature and abandon the Texas State Capitol inflicts harm on the very Texans who elected them to serve,” he said. “The Democrats must put aside partisan political games and get back to the job they were elected to do.”

Both the House and Senate held public hearings on a set of new bills Saturday, hearing testimony that stretched overnight until early Sunday morning. Both committees advanced the bills, and one or both of them could have seen a floor vote by the full body as soon as Tuesday.

Bills filed during the special session by both the state House and Senate would limit early voting hours, place additional restrictions on assisting disabled voters and voting by mail, and make many election missteps felony offenses. The special session was also set to consider funding border security measures, banning the delivery of abortion pills and banning transgender students from competing in university sports in their non-birth gender. It was also set to reinstate the pay of legislative staffers, which Mr. Abbott vetoed after the May walkout.

The Texas Public Policy Foundation, a conservative think tank, posted on its website a poll Monday saying that 54% of Texans oppose members staging walkouts. “Most Texans see it as a childish and desperate move, and they don’t like temper tantrums,” the group’s spokesman Brian Phillips said in a statement.

Carisa Lopez, political director of Texas Freedom Network, a civil liberties nonprofit, said lawmakers told her they are prepared to stay out of the state as long as necessary, despite the hardship. “We’re in the fight of our lives here, and they know that,” she said.

Ms. Lopez said the decision to leave the state was in part an effort to help national voting legislation move forward. Last month in Washington, Senate Republicans blocked Democrats from debating election legislation put forward by Democrats. The legislation has been a priority for Democrats, who say it would preserve and expand voter access at a time when many GOP-led states are pursuing more voting restrictions. Republicans have called the Democrats’ push a power grab that would undercut election security.

The Democrats’ flight to Washington will likely not succeed in killing the Texas Republicans’ election bill, but it will help Democrats spread their message on a national level, said Cal Jillson, a Texas political expert and professor of political science at Southern Methodist University in Dallas.

Mr. Jillson said that it would be difficult for Democrats to wait out almost an entire month of the legislative session out of state, and even more difficult to do so again if Mr. Abbott calls additional sessions.

“What the Democrats are doing here is saying, ‘We’re getting beat down here in Texas, so either we accept the defeat or extend the scope of the fight—take it national—and we may still not win the fight, but we might win it rhetorically,” Mr. Jillson said. “Why the hell not?”

National attention has focused on Texas election legislation after Republicans in Florida and Georgia also passed bills to tighten voting procedures. Florida’s law, signed by Gov. Ron DeSantis in May, included new rules governing drop boxes, new identification requirements for voters seeking mail ballots and prohibitions on activity that could influence voters within 150 feet of a polling place or drop box. In Georgia, measures passed in March included new identification requirements for mail-in voters and limits on ballot drop boxes.

Democrats began fundraising off the walkout as it was under way Monday. Former U.S. Rep. Beto O’Rourke encouraged followers on Twitter to “Support Texas House Democrats as they fight back against voter suppression in Texas and take the fight to our nation’s capitol.” A linked fundraising page said donations would help provide resources to block Texas voting bills.

In Philadelphia on Tuesday, President Biden will make the case to the public for federal legislation to curb GOP-led efforts in states to pass more restrictive election laws, said White House press secretary Jen Psaki.

“Part of what you need to do is build a grass-roots army across the country to engage with, to empower, to inform about what their rights are and what the challenges are,” she said.

Ms. Psaki said the president doesn’t accept that voting rights-related legislation is dead in Congress and wants lawmakers to find a path forward, despite GOP opposition in the Senate.

Ms. Psaki said as of Monday the president had no plans to meet with Texas Democrats.

Texas House Democrats made a similar move in 2003, when dozens fled to Oklahoma to block a Republican-drawn redistricting plan that would have meant a loss of four or five Democratic seats. More than 50 Democratic lawmakers fled the state at the time in buses, where they would be less likely to be pursued by Texas Rangers and the Department of Public Safety.

State senators then broke quorum during a subsequently called special session and legal battles ensued. Before that, Democrats last tried to bust a quorum in 1979, when a dozen state senators, dubbed the “Killer Bees” left the state to stop a bid to separate party primary days.

—Alex Leary and Tarini Parti contributed to this article.

Write to Elizabeth Findell at Elizabeth.Findell@wsj.com