"We're going to present, after we get sort of buy-in from our Republican Conference, to our Democratic colleagues to say this is a simple, straightforward solution that doesn't carry with it a lot of the baggage that usually drags down immigration reform", the Texas Republican told reporters Tuesday.
Trump reportedly urged House Republicans to get an immigration bill to his desk soon, vowing to stand with them "1,000 percent" as they attempt to pass legislation meeting his demands.
And it was uncertain if the President's support would be enough to push any legislation through the divided Republican majority. "They're in schools. They're in the police", she said.
"We want to end the border crisis by finally giving us the legal authorities and the resources to detain and remove illegal immigrant families all together and bring them back to their country", he said.
While top officials have also stood by Trump's "zero tolerance" approach, insisting children are being held in humane conditions, criticism has swelled from global rights groups, Christian evangelicals, former U.S. first ladies and the president's own Republican Party.
The UN High Commissioner for Refugees, Filippo Grandi, has told the BBC the Trump administration's "zero-tolerance" policy towards illegal immigration is unacceptable.
The Republican president blamed Democrats for not coming to the table to negotiate immigration legislation.
Both the House and the Senate are controlled by Republicans. If separations occur, Homeland Security would have to provide clear guidelines for how parents can contact their kids.
Almost 2,000 children were separated from their parents between mid-April and the end of May.
Trump insisted he had no choice but to criminally prosecute every adult crossing into the United States from Mexico, meaning "you have to take children away".
It requires the government to release rather than detain children caught trying to enter the USA and exempts them from an immediate return to their home countries. There are 100 sites scattered across 17 states and they can be on the other side of the country from their parents.
Warren also criticized the White House for not taking action to stop the controversial family separations.
The major change being unveiled Tuesday would loosen rules that now limit the amount of time minors can be held to 20 days, according to a GOP source familiar with the measure.
Photos and video have shown children in cages or without beds as they beg for their parents. Senate leaders are working through various proposals to find consensus as more and more Republicans are speaking out in sharp opposition to Trump. Cornyn said they want to keep families together while still "expediting their ability" to go before an immigration judge.
"We had a great meeting", Mr Trump said after meeting Republican lawmakers on Capitol Hill on Tuesday night.
"While cases are pending, families should stay together", tweeted Senator Cruz, who is in an unexpectedly tough re-election battle. Earlier, the proceedings of migrant families were held in a civil court so that the families can be held together.
As for Congress, Hurd was dismayed that legislation is needed at all.
"The president is trying set this trap in the public mind that somehow there is a law requiring him to do this and Congress can undo it", said Senator Chris Van Hollen, who visited a detention centre in Brownsville, Texas, over the weekend.
"You saw the President [say] on camera that he wants this to end".
First of all, there is absolutely zero correlation between Germany's immigration policies and crime.
To prosecute illegal migrants in the U.S.
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