Pope Francis also joined the public outcry, telling Reuters he agreed with US Catholic bishops that separating children from their parents at the border was "immoral".
"There will not be a grandfathering of existing cases", Kenneth Wolfe, spokesman for the Administration for Children and Families, part of HHS, told The Times.
The House of Representatives planned to vote on Thursday on two bills created to halt the practice of separating families and to address other immigration issues.
US officials say more than 2,300 children have been separated from their parents or guardians since early May, when the administration announced its push to arrest and charge anyone illegally crossing the US-Mexico border, regardless of whether they were seeking asylum. Under the Obama administration, such families were usually referred for civil deportation proceedings, not requiring separation.
Homeland Security Secretary Kirstjen Nielsen reportedly drafted an executive action and took it to the White House, which would instruct the Department of Homeland Security to keep families together. One official said it was the Justice Department that generated the legal strategy that is codified in the executive order, and disputed the notion that Homeland Security was involved in drawing up the document.
Trump said his immigration policy "continues to be zero tolerance".
Katharina Obser, a senior policy adviser at the Women's Refugee Commission, said the order could result in expanded detention for immigrant families.
The policy stemmed from the zero-tolerance prosecution of illegal immigrants, put in place some months ago by the U.S. Attorney General, Jeff Sessions. "Legislation is not the way to go here when it's so easy for the president to sign it". "We want to keep families together", he said Wednesday.
Trump told Republicans during a closed-door meeting Tuesday night that he would support them "1,000 percent" on immigration but did not specifically back either bill.
"They really would like to have open borders where they can just flow in", Mr. Trump said of congressional Democrats. They crossed through Mexico to reach the USA border. "And children shouldn't be punished for their parents breaking the law".