Congressional Republicans are racing to find legislative options to stop a White House policy of separating migrant children from their parents at the U.S. border amid widespread condemnation of the practice.

Republicans in both the House and the Senate began ramping up pressure on the White House to reverse the policy Monday amid outcry about children being housed in detention centers after being separated from their parents. President Trump is scheduled to meet with House Republicans on immigration Tuesday evening and lawmakers expect the family separation issue be a prominent part of that conversation.

Administration officials have insisted that Congress is responsible for changing the law, even as many lawmakers argue that Trump has the power to simply choose to enforce his policy differently. Utah GOP Sen. Orrin Hatch is among those who have strongly insisted that the Administration resolve the issue on its own.

"I think the White House can fix it if they want to," Hatch said Monday. "I don't think there's any question about that."

The fight is adding even greater pressure on Congressional Republican to pass legislation despite deep, long-standing divides within the party over how to handle both legal and illegal immigration.

The House is expected to vote later this week on a pair of immigration proposals that are expected to address the issue of family separation but it isn't clear that either bill has the votes to pass. Many Republicans say privately that they are concerned that a failure to act quickly will result in a prolonged crisis and public outrage.

The result has been a haphazard series of targeted legislative proposals from different factions of the GOP that address pieces of the problem, instead of a broader immigration reform effort.

Texas Sen. Ted Cruz was among the first Republicans to say he will introduce legislation which would double the number of federal immigration judges to 750, expand detention facilities with temporary shelters and expedite processing for asylum claims while mandating that families be housed together during the proceedings.

"We should fix the backlog in immigration cases, remove the legal barriers to swift processing, and resolve asylum cases on an expedited basis," Cruz said in a statement accompanying the release of the bill's outline. "While these cases are pending, families should stay together."

The Cruz proposal has yet to receive any strong support within the Senate but GOP leaders are hoping to craft a consensus plan in the coming days, according to GOP sources. Other Senators, including Sens. Lindsey Graham, R-S.C., and John Cornyn, R-Texas, also say they are working on legislation.

Cornyn, the number two Republican in the Senate, in particular has great sway within Senate GOP circles, and is working with Rep. Henry Cuellar, D-Texas, on a bill to expedite hearings and keep families together.

"The question is is Congress going to step up and deal with this unfortunate by product where families are separated as a result of court decisions, consent decrees and other laws that are within our power to correct," Cornyn told reporters Monday night. "I think we all are concerned about making sure these children are treated in a compassionate and humane way. That is what we would like to do but if people won't work together to do it then our hands are really tied."

Cornyn was among the first Senators to agree with the White House position that Congress has a responsibility to address the issue.

Another proposal from conservative House Freedom Caucus Chairman Rep. Mark Meadows, R-N.C., would keep families together while cracking down on the rules for those seeking asylum. Meadows announced the proposal at an informal press conference at the White House on Tuesday following a meeting with Trump.

"By cutting down on asylum fraud, we can keep families together without running a greater risk of traffickers or violent criminals using children to take advantage of the rules," Meadows said in a statement. "We need to better enforce our immigration laws, but we can do so while keeping parents and children together."

Meadows said his bill is a backup plan if the House fails to pass broader immigration proposals this week.

Democrats are not backing any of the Republican efforts to end family separation. House Minority Whip Steny Hoyer dismissed language in the broader House Republican immigration bill, saying it "doesn't deal with the issue." He said Democrats want to see an up-or-down vote on a stand-alone bill along the lines of the measure introduced by California Sen. Diane Feinstein, and supported by all 49 Senate Democrats.

Feinstein's proposal hasn't gained traction among most Republicans, in part because it includes pointed language some view as political.

"An agency may not remove a child from a parent or legal guardian solely for the policy goal of deterring individuals from migrating to the United States," the bill says. "Or for the policy goal of promoting compliance with civil immigration laws."

Top Republicans have dismissed that bill as insufficient.

"It doesn't solve the real problem," Cornyn said.

But Colorado GOP Rep. Mike Coffman, who is facing pressure to act as one of the most vulnerable House GOP members facing re-election this fall, tweeted that he was supportive of the the Democratic bill.

House Republican leaders are primarily still focused on upcoming votes on a pair of broader immigration bills. The first bill up for a vote on Thursday will be a hard-line conservative proposal written by House Judiciary Committee Chairman Bob Goodlatte, R-Va., that is expected to fail. The second bill is a compromise between conservatives and centrists that has been brokered by House GOP leaders. That bill has a slightly better odds, though leaders have stopped short of promising it will pass.

Support for the compromise bill may hinge on what Trump tells House GOP members at the closed-door meeting Tuesday night. Many members are wary of supporting any immigration bill following a chaotic week where Trump appeared to switch his position in immigration several times.

Some members fear that Trump has a history of mercurial support for policies, like the omnibus spending bill earlier this year and during the health care debate in 2017. In both cases Trump said he backed certain polices only to turn around later and blast the same ideas--and those who voted for them--later on Twitter.

That lack of confidence and fear of retribution could make it significantly harder for Republicans to pass any politically risky immigration proposals ahead of the midterm elections in November.

NPR's Scott Detrow contributed to this report

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