The senior administration official who briefed reporters even as Mr. Trump was speaking said that while unilateral family separations are off the table, a policy dubbed "binary choice" is still being considered.
After the judge's ruling, Mr Trump said on Twitter the decision was "unfair to the US" and "out of control".
While any appeal against Judge Seeborg's ruling will be heard by the federal Ninth Circuit appeals court - a frequent target of Mr. Trump's ire - Judge Seeborg himself is not on the circuit.
Vice President Mike Pence plans to visit the US Border Patrol Station in Nogales, Ariz., on Thursday, where he will be briefed by Customs and Border Protection agents and tour the border wall, according to a White House official.
"Many, many dead people", Trump said, referring to migrants who he said had died along the journey.
Last week, Trump threatened to impose steep import tariffs on Mexican automobiles if Mexico does not do more to stop would-be migrants on their trek north to the USA border. "We found cases where adults have been coming in with kids who are not their kids, yet they're claiming that they're their children".More news: Magic Johnson Steps Down as Lakers President
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Immigration court records show the asylum denial rate rose from 21 percent in 2016, the previous year of the Obama administration, to almost 47 percent as of January. "They have all these disgusting laws that the Democrats won't change they will not change them".
Law enforcement agents apprehended or turned back 103,492 migrants attempting to reach the USA southern border in March, marking the highest month in 12 years.
Judge Richard Seeborg has given the government until the end of the week to seek a stay of his Monday ruling in which he said he would grant a request by civil liberties groups to halt the practice while their lawsuit proceeds. The White House did not immediately respond to questions seeking clarity on the commander in chief's comments.
He explained that his order does not decide the issue of whether Congress could authorize the Department of Homeland Security to create such a program, whether DHS could implement MPP if it had more safeguards or whether it is a "wise, intelligent or humane policy".
But the official told the New York Times that there was room to tighten the criteria for deciding whether the claim was plausible, and reject more asylum seekers during the screening process.
In addition, around 10,800 people who presented themselves at ports of entry were deemed inadmissible in March, bringing the number of total border enforcement actions to above 103,000, the most for one month in more than a decade.
Some asylum-seekers affected by the program have told the immigration judge assigned to their cases how much more hard being in Mexico has made navigating the court system.