The Trump administration argues that the Flores settlement means children must be separated from their parents because they can not be held in custody with their parents who face prosecution for crossing the border illegally.
Contrary to the president's public statements, no law requires families to necessarily be separated at the border.
But a backlash is mounting, fuelled by reports of children being taken from mothers and distraught toddlers and young children asking, through tears, when they can see their parents.
Democrats have accused Mr Trump of effectively turning the children into political hostages to secure stricter immigration measures, such as funding for a US-Mexico border wall.
Sessions and officials from the Department of Homeland Security have defended the practice of separately detaining children and parents trying to cross the border, which has led to about 2,000 children being separated from their parents in the past 45 days.
"House Republicans are not going to take on immigration without the support and endorsement of President Trump", said Rep. Patrick McHenry, R-N.C., the GOP's chief deputy vote counter.More news: Strong natural disaster hits Osaka in western Japan leaving people dead
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In the meantime, Donald Trump is holding USA immigration policy hostage in the hopes of funding a border wall that recent polls show nearly 70% of Americans don't want to fund with taxpayer dollars.
He will meet with Republican colleagues on Tuesday to discuss both bills, which the White House has said he would be willing to sign.
Trump adviser Kellyanne Conway said Sunday on NBC's "Meet the Press" that she "very forcefully" objected to the implication that Trump sought to use immigrant children as a bargaining chip, even though the president himself suggested it.
In a tweet on Saturday he urged Democrats to work with Republicans to create new legislation.
Stephen Bannon, who helped guide President Donald Trump's 2016 campaign to victory and then served briefly as top White House strategist, said Sunday that the president should draw a line in the sand over his proposed border wall.
HHS is tasked with caring for "unaccompanied alien children", which by law is any minor that arrives at the border without parents or relatives, or for whom loved ones were not "available to provide care and physical custody".
"This is inhumane. I'd like to say it's un-American, but it's happening right now in America", O'Rourke added.
"The Democrats are forcing the breakup of families at the Border with their disgusting and cruel legislative agenda", the president tweeted on Friday, again blaming Democrats. "The Democrats have to change their law - that's their law".
Ms Conway, however, put the onus on Democrats, saying if they were serious about overhauling the system, "they'll come together again and try to close these loopholes and get real immigration reform". We want you to imagine for a moment what this might be like for a child: to flee the place you have called your home because it is not safe to say and then embark on a unsafe journey to an unknown destination, only to be ripped apart from your sole sense of security with no understanding of what just happened to you or if you will ever see your family again.