Trump's policy took children away from parents who were apprehended crossing the border illegally.
He bowed to public pressure and reversed the policy earlier this week.
President Barack Obama did not oversee the separation of 90,000 migrant children and their parents at the USA border, contrary to a misleading online report and claims circulating on social media.
The father told the AP in a phone interview that he spoke to his son once by phone since they were separated 26 days earlier.
An ICE official said it is unclear how families will be reunified.
"These are the American citizens permanently separated from their loved ones", he said.
"We must maintain a Strong Southern Border", the president tweeted.
"We'll not rest until our border is secure, our citizens are safe and we end this immigration crisis once and for all", Mr Trump added.
Laura Wilkerson's 18-year-old son was murdered in 2010.
The Department of Defense is ready to make space available on military bases for as many as 20,000 unaccompanied migrant children, a spokesman told the Associated Press.
They were trying to persuade colleagues to seize the moment and tackle immigration problems by approving the bill, which includes $25 billion for Trump's border wall and a path to citizenship for young immigrants who have lived in the USA illegally since childhood.
Doubts remain over the logistics of reuniting them all, however, with some parents having been unable to get in touch with their children as they are being held in different containment camps hundreds of miles away.
Images of children in metal cages, and audio of young migrants crying out for their parents sparked protests and drew worldwide condemnation from world leaders and organisations.
"We weren't lucky enough to be separated for five days or 10 days, we're separated permanently", she said.
After a week on his heels, Donald Trump has begun his counterattack.
The bill set to be voted on next week would provide US$25 billion for Trump's long-sought border wall, offer a pathway to citizenship to young undocumented immigrants known as "dreamers" and keep migrant families together in detention centres.
On Friday afternoon, the president stood alongside relatives of those killed by undocumented immigrants. Several of those speaking condemned the media for ignoring the stories of the victims and praised Trump and Vice President Mike Pence for their attention to border security. These were families that had been "permanently separated".
Trump backed up his complaint with the unsupported claim that undocumented immigrants are more unsafe than the general population and shared statistics on crimes committed by undocumented immigrants without any context.
Democratic lawmakers said they aren't convinced that the government has any real plans to reunite the families.
"Seventy-five percent (75 percent) of Democrats think the Trump administration is too aggressive in trying to stop the flow of illegal immigrants into the United States - a view shared by only 23 percent of Republicans and a plurality (46 percent) of unaffiliated voters".
Despite the GOP leadership's vows to fight on, some Republicans were openly skeptical about their ability to achieve success in the wake of Trump's latest tweets. But the bill was delayed to Friday before being pushed back to next week.
Rather than negotiate with congressional Democrats on immigration legislation, President Donald Trump urged his fellow Republicans to abandon the issue until next year.