Trump is planning to sign an executive order to allow children to stay with parents caught crossing the border illegally - a step that could avoid the family separations, Fox News' John Roberts reported.
The White House may run into legal difficulties if the executive order requires families to be held intact together indefinitely since a federal consent decree known as the Flores settlement says children must be released after 20 days.
Trump, addressing what his administration has characterised as an unwanted side effect of his zero-tolerance policy on illegal immigration, signed the order to keep families who illegally cross the U.S. southern border together as they await immigration proceedings.
"We're going to have strong, very strong borders, but we're going to keep the families together", said Trump, who added that he didn't like the "sight" or "feeling" of children separated from their parents.
The Trump administration has come under attack by religious leaders, Democrats and Republicans as reports surfaced that more than 2,000 children have been separated from their families since the Department of Homeland Security initiated a "zero tolerance" policy.
The president and his top aides have insisted for days that they had no choice but to separate children from their parents due to unspecified United States laws. USA protocol prohibits detaining children with their parents because the children are not charged with a crime and the parents are.
Perry was not at the meeting with Trump, but said he doubts the president's words will affect his position.
Their cases usually take years to resolve because of the big backlog in immigration courts, where judges decide whether the children can stay in the US or must be returned to their home countries.
But after signalling Monday that it would oppose any fix aimed exclusively at addressing that issue, the White House said Tuesday it was reviewing the emergency legislation being introduced by Cruz to keep migrant families together.
Because any illegal border crossing is prosecuted, parents and children are separated during the legal process. We, as a nation, have a moral obligation to assist these families.
People protest against the Trump administration policy of separating immigrant families suspected of illegal entry, in New York, June 19, 2018.
Michigan's Department of Civil Rights says immigrant children as young as three months of age have arrived in the state for temporary foster care placement after being separated from their parents at the US border.
Sen. Chuck Schumer, the Democratic leader from NY, is adamant that Trump can end the family separations on his own and that legislation is not needed.
House Judiciary Committee Chairman Bob Goodlatte's bill, the Securing America's Future Act, provides a pathway to legal status for Dreamers while limiting legal immigration levels. The new Trump policy, unveiled Wednesday, involves keeping families detained together for an indefinite amount of time - a time that could stretch into months or even years if that family is seeking asylum. "This is not something that happened just now", he said.
The order aims to end the separation of most migrant families by detaining them together while they await their court dates in criminal and immigration court.
Senate Republicans were preparing to advance legislation that would prohibit the government from splitting up families apprehended after crossing the border.
Immigrant parents are much more likely to find their child or children if they have attorneys or advocates to track them down.
President Trump met with House Republicans to discuss immigration policy this afternoon in an atmosphere that was approaching a real political crisis.