The administration detained families together, sometimes indefinitely, in an attempt to deter more illegal immigration.
Trump continued to blame opposition Democrats and the media for the crisis on the southern USA border.
It was not immediately clear what Trump, who had previously blamed the family separations on Democrats, would sign.
In his Wednesday statement, the former president suggested Americans should hold their lawmakers accountable to such promises. "We've got to have the rule of law in this country". Trump has also claimed that he's merely enforcing an Obama-era policy, but it is his White House policies that are criminalising border crossings to this extent and triggering family break-ups.
Parents who are referred by border agents for prosecution are held in federal jails, while their children remain in US Customs and Border Protection custody or are moved into detention facilities managed by the Office of Refugee Resettlement, a Department of Health and Human Services agency.
Both bills include language to end the controversial White House policy of separating migrant parents from their children at the border.
The revision would also give the Department of Homeland Security the authority to use $7 billion in border technology funding to pay for family detention centers, said the person, who was not authorized to discuss the matter by name and commented only on condition of anonymity.
Without Democratic support, Republicans can not muster the 60 votes needed to move forward on legislation.
"Put simply, the deliberate separation of children and their parents who seek lawful asylum in America is wrong", the letter from the attorneys general reads, adding that the practice is "contrary to American values".
Trump: Will sign ‘something’ to keep families together
The Trump administration says parents who present themselves at official ports of entry along the worldwide border and seek asylum won't be prosecuted and won't be separated from their kids. The government says that only happens when immigrants can not prove family relationship or when there is evidence children have been abused. "Republicans want security. But I am working on something - it never ends!" A particularly harrowing account told of parents being informed that their child was being taken away for a bath - only to be told half an hour later that they would not see them again.
Bob Goodlatte, R-Va., chairman of the House Judiciary Committee, and a compromise immigration bill.
On Tuesday, Trump was greeted with jeers from some Democratic members of Congress as he left a session with House Republicans on immigration policy.
He ends his message with a call to action, pushing Congress to take legislative action and all Americans to vote.
"It's incredible how people are surprised that he's keeping the promises he made on the campaign trail now", said Trump political adviser Bill Stepien. "I mean it is hard for me to even wrap my mind around it", said Kay Bellor, vice president for programs at Lutheran Immigration and Refugee Service, which provides foster care and other child welfare services to migrant children.
A potential executive order could emerge as a fallback option for the administration if it wants to alleviate the crisis should the House bill not pass - but talk that it is a possibility seems likely to undermine momentum for Republican senators to take a tough immigration vote.
A White House official said first lady Melania Trump had been making her opinion known to the president for some time that she felt he needed to do all he could to help families stay together, whether by working with Congress or acting on his own.More news: First case of West Niles reported in IL in 2018