The Texas Republican said his bill emphasizes the "four pillars" of border security, ending chain migrations, discontinuing the visa lottery and taking a merit-based approach to the DACA recipients who were brought here illegally as children.
One woman, Cindy Madrid from El Salvador, repeatedly dictated her US-resident sister's phone number to her six-year-old daughter before she crossed the border and the family was separated.
United States president Donald Trump may have doomed the chances for a House Republican immigration bill after urging party lawmakers to abandon the compromise effort they have been working on for weeks, the media reported.
In McAllen, the visiting lawmakers said they believed border agents were handling the situation as well as could be expected at the facility for immigrants recently apprehended along the border. Senior officials are crafting a plan for reuniting immigrant children with their parents or guardians, though it remained unclear how long that work will take.
Last week, Trump reversed a policy of separating families entering the US illegally at the border with Mexico.
President Donald Trump has tried to cast doubt on wrenching tales of migrant children separated from their families at the border.
But stubborn differences between conservative and more moderate Republicans in the House have stalled immigration legislation on Capitol Hill, with a vote on a compromise measure delayed until next week.
To fellow Republicans, his message was to "stop wasting their time on Immigration" until after the November mid-term congressional elections.
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An administration official said on Friday evening that U.S. Customs and Border Protection "expects that all unaccompanied children in their custody who were separated from adults who were being prosecuted will have been reunited with their families".
Democratic lawmakers on Saturday said they aren't convinced the Trump administration had any real plan to reunite immigrant families caught along the southwest border, while demonstrators gathered to protest the separation of parents from their children by USA border authorities.
"We can not allow our Country to be overrun by illegal immigrants as the Democrats tell their phoney stories of sadness and grief, hoping it will help them in the elections", he tweeted. He has falsely blamed Democrats for forcing them to enforce the policy.
"Game over", said Representative Mark Sanford, a Republican critic of Trump who came under withering attack from the president before losing his primary race this month.
Rep. Anna Eshoo, D-Calif. said the current policy of separating kids from their parents has "brought great shame to a great country".
His proposal drew immediate criticism from legal analysts and immigrant rights advocates who said it would violate the U.S. Constitution's due process provision, which applies to citizens and non-citizens alike. They also downplayed the potential impact of Trump's tweet. The immigrants are mostly teenage boys and housed in tent-like structures. This comes as a result of the White House's "zero tolerance" policy toward illegal border crossings.
The Trump administration called last year for the expansion of the expedited removals programme to immigrants who have been in the country illegally for up to two years.