Updated at 5:40 p.m. with comment from Sen. Ted Cruz and Rep. Bill Flores.

WASHINGTON — As the White House begins winding down an Obama-era program that shields children brought to the country illegally from deportation, some Texas lawmakers — both Republicans and Democrats — are calling for Congress to act on a long-term fix.

But it’s too soon to know what that solution will look like, whether it will gain the broad support of both GOP-led chambers, and if lawmakers — already tasked with high-profile to-dos, such as rewriting the nation’s tax code — will pass new legislation before major changes take effect in March.

A congressional overhaul of the Deferred Action for Childhood Arrivals program is seen as a long shot given historical rifts over immigration, a task made even more complicated by talk that the White House could favor tying the measure to funding for President Trump’s pledged border wall.

"It now falls upon Congress to take swift and immediate corrective action to protect these young men and women who are openly contributing to the fabric of this great nation. Dreamers need to know they are safe," said Houston Rep. Sheila Jackson Lee, a Democrat who called on Congress to enact an "emergency DACA protection provision" in the upcoming continuing resolution to keep the government funded.

Other Democrats urged a vote on a bipartisan version of the Dream Act, which would grant permanent legal status to certain immigrants brought to the U.S. as minors.

But the Dreamers' fates largely rest on Republican support. And Trump’s decision hands his party, deeply divided on immigration issues, what could be its latest high-profile conflict, while signaling he'll take it up in the future if they fail to act.

Congress, get ready to do your job - DACA! — Donald J. Trump (@realDonaldTrump) September 5, 2017

Congress now has 6 months to legalize DACA (something the Obama Administration was unable to do). If they can't, I will revisit this issue! — Donald J. Trump (@realDonaldTrump) September 6, 2017

GOP divisions

Several Texas GOP members applauded Trump's decision but were short on details about what they'd like to see next, including Round Rock Rep. John Carter and San Antonio Rep. Lamar Smith, an immigration hardliner who said Congress should focus on ensuring "that our immigration policies put unemployed Americans first."

And though Sen. Ted Cruz lauded Trump for "acting to fulfill his constitutional duty" by rescinding the program, he avoided repeated questions about whether he supports allowing DACA recipients to be granted legal protection.

But San Antonio Rep. Will Hurd, a Republican who represents a swing district in West Texas, said Congress should provide a “permanent, legislative solution” for the program that shields roughly 800,000 immigrants from deportation, including roughly 124,000 in Texas.

So did Bryan Rep. Bill Flores, who praised Trump for rescinding the program but wants the Dreamers to be legally authorized to stay.

"When it comes to the Dreamers, those children and young adults who are here through no fault of their own, I believe Congress should quickly find a solution to ensure they can stay in the United States, which for many is the only home they know," he said in a statement.

In an interview with The Dallas Morning News, Dallas Rep. Pete Sessions, chairman of the House Rules Committee, said "Congress should accept this leadership challenge to work with the president," adding: "I will do my very best to make sure that a piece of legislation addressing this, my name is on."

Asked whether he supports allowing DACA recipients to remain in the U.S., Sessions noted he has long favored a guest-worker plan that would take them into account. On Monday, he told CNN that granting legal status as a guest worker is the "minimum" Congress should do.

Sen. John Cornyn, the No. 2 Republican in the chamber, is also calling for Congress to find a “long-term resolution" to the DACA issue, noting: “I don’t know anybody who believes that children brought to the country underage are somehow culpable or responsible legally, so I would support a solution there."

It will fall to the Texas senator, who just spent months unsuccessfully trying to wrangle votes needed to dismantle the Affordable Care Act, to secure support needed for a DACA solution in the Senate.

He’s up against a divided GOP conference already considering drastic changes to the nation’s immigration laws. Earlier this year, Sens. Tom Cotton of Arkansas and David Perdue of Georgia introduced a bill that aims to slash legal immigration by half over a decade.

Arizona Sen. Jeff Flake said he wants Congress to pass standalone legislation to “lawfully ensure” children brought here illegally can remain, though the White House wouldn’t commit to that on Tuesday.

Others, including Sens. Pat Toomey, R-Pa., and Ron Johnson, R-Wis., indicated a desire to tie the DACA issue to other immigration-related measures, such as border security and cracking down on companies that hire illegal immigrants.

Bipartisan solution?

The move to phase out DACA is giving new attention to legislation already under consideration in the House and Senate.

Chief among them is a bipartisan measure introduced by Sen. Lindsey Graham, R-S.C., and Sen. Dick Durbin, D-Ill., that would grant permanent legal status to more than 1 million young people who arrived in the country before age 18, and who passed security checks and met other criteria, such as joining the military, according to the Associated Press.

Some high-ranking Democrats are calling for the Senate to take up the bipartisan bill, including California Sen. Dianne Feinstein, the top Democrat on the Senate Judiciary Committee, and House Minority Leader Nancy Pelosi.

Dallas Rep. Eddie Bernice Johnson has also joined House Democratic leaders in calling for Congress to take up the measure. “This bill should be brought to the floor without delay because DACA program recipients should not live in fear and uncertainty of their future for a path towards legal citizenship,” she said in a statement on Tuesday.

But Cornyn indicated the bipartisan bill is unlikely to advance “by itself,” suggesting it could be wrapped into a larger, more comprehensive immigration package.

“There’s going to be a lot of debate about what that will look like and what ought to be added to it,” he said, later telling Politico there's "no way" a DACA fix will pass as a standalone measure.

In a news conference with Durbin on Tuesday, Graham predicted Republicans would support his bill, if Trump backed it. Doing so, he said, would help demonstrate that the GOP both cares about following the constitution but has compassion for children brought here illegally.

"To the president, you have a chance to show the nation, as the president of all of us, where your heart is," Graham said.

"To the president, you have a chance to show the nation, as the president of all of us, where your heart is at," Sen. Lindsey Graham says. pic.twitter.com/VfOz5hoX1l — CBS News (@CBSNews) September 5, 2017

The Graham-Durbin measure is one of a handful of bills under consideration in both chambers.

Colorado Rep. Mike Coffman, a Republican, is expected to file a discharge petition that would force the House to vote on his “Bridge Act,” legislation that would extend DACA for three years, according to NPR.

Staff writer Nicole Cobler contributed to this report.