Gov. Greg Abbott of Texas on Thursday called a new special session of the Legislature that is set to begin on Saturday, renewing Republican efforts to overhaul the state’s elections and putting pressure on Democratic lawmakers who left the state for Washington last month to block the legislation.

Mr. Abbott, a Republican, stuck to his pledge “to call special session after special session,” releasing a 17-item agenda for the Republican-controlled Legislature with a new voting bill at the top. The list also included a host of other conservative goals, like restricting abortion access, limiting the ways that students are taught about racism and tightening border security.

His announcement sent national attention swinging back to a hotel in downtown Washington, where several dozen Democrats from the Texas House of Representatives are grappling with a familiar question: Stay or go back?

The Texas Democrats are torn over how much is left for them to accomplish in Washington, with some moderate members of the caucus believing that their point has been made. But more progressive members are pushing to stay in Washington and continue to call attention to voting rights, at least while the U.S. Senate remains in session.

“I’ve been very clear, as it relates to me, that as long as Congress is in town, working on voting rights, I will be here in Washington, D.C., advocating for voting rights,” said State Representative Trey Martinez Fischer, a Democrat who was one of the organizers of the initial flight from Austin.

President Biden’s administration, by contrast, appeared to suggest that it would support a return to Texas by the state lawmakers.

“Certainly, the president believes that, one, they’ve been outspoken advocates and champions of voting rights,” Jen Psaki, the White House press secretary, said at a news conference, adding that if the legislative calendar “required them to be there, we would support that.”

The lawmakers’ stay in Washington has amounted to a prolonged period of limbo; their trip has delayed Republicans’ attempt to pass an election bill, but it remains unlikely that it will be a fatal blow.

Federal officials celebrated their arrival in Washington, with Vice President Kamala Harris likening their departure from Texas to the voting rights march in Selma, Ala., and other famous civil rights protests of the 1960s. But the group lost momentum when several vaccinated legislators tested positive for the coronavirus.

In video chats, the Texas Democrats did their best to maintain pressure on both the White House and Democratic senators to find a path forward for federal voting legislation, and eventually coaxed more than 100 state legislators from other states to join them in Washington.

And the lawmakers’ visit to Washington has coincided with the renewal of talks toward a compromise voting bill. Eight Democratic senators, including Senator Joe Manchin III of West Virginia, have been moving closer to a final draft to be introduced later this year. What prompted the end of congressional inertia, however, is unclear, and any federal voting bill would remain unlikely to move quickly through the chambers of Congress.

So now, with the Texas Democrats confronting an uncertain future, they are debating their next moves.

If they return, they could be subject to the as-yet-untested powers of the Republican Statehouse leadership to arrest and detain any lawmakers who do not show up for a legislative session while in the state of Texas.

While Speaker Dade Phelan, a Republican, can issue arrest warrants during a session that has been gaveled in, there has never been a test of that authority when a session has been called by the governor but cannot start because enough lawmakers have declined to show up. Mr. Phelan’s office believes he has the authority to request arrest warrants and send law enforcement officers to retrieve absentee lawmakers even if the session has not started.

Back in Austin, Republican members said they had been maintaining informal discussions with their Democratic colleagues in an attempt to re-establish a quorum and get back to work. The partisan strictures in the Texas Legislature are far less rigid than those in Congress, with no dividing aisle between Republicans and Democrats. Members of the opposing parties intermingle more on the House floor and often form working friendships.

“I can tell you they’ve been going on since they left three weeks ago,” State Representative Jim Murphy of Houston, the chairman of the 83-member House Republican Caucus, said of the largely ad hoc discussions. Most of the conversations were “just personal — largely people want to know if they’re going to return,” he added. “How committed are they? Are there some that are willing to come back? Are there things that need to happen to encourage them to return?”

“I’ve done some texting, some phone calling,” he said, though “not a whole lot.”

At least nine Democrats have remained in Austin for varying reasons, though most, if not all, have embraced their colleagues’ opposition to the voting bill.

But as Democrats consider their immediate future, Mr. Abbott did add a surprise item to the agenda that, while unclear in its scope or likelihood of success, could further complicate their calculations: “Legislation relating to legislative quorum requirements.”

Katie Rogers contributed reporting.