The US political battle over migrant children separated from parents has shifted to Congress, after Donald Trump signed an order to halt the policy.
The House of Representatives will vote on broader immigration bills that would fund President Trump's proposed border wall and reduce legal migration.
But Democrats oppose the measures and it is uncertain they will pass.
Mr Trump reversed his policy after an outcry at home and globally over the policy.
The Republican president's executive order calls for the families to be detained together while their cases are considered.
But the fate of families already separated by the policy is not clear.
At least 2,300 children have been taken from more than 2,200 parents since 5 May, according to US immigration officials.
Mr Trump has previously said it was up to Congress to resolve the issue, in combination with tougher immigration regulations.
One of the House bills is a compromise between conservatives and moderates. It includes funding for a border wall, along with a pathway to citizenship for the Dreamers - immigrants brought illegally into the US as children.
"This is a bill that has consensus. This is a bill that the president supports. It's a bill that could become law," said House Republican leader Kevin McCarthy, quoted by the Associated Press.
The second, more conservative bill excludes the Dreamers clause.
With no Democratic support for the measures, White House officials have been seeking to drum up support among Republicans, many of whom are wavering or openly opposed.
Meanwhile, the Senate is working on a narrower compromise bill focusing on the family separations.
The president signed his executive order on Wednesday evening.
"I did not like the sight of families being separated," he said, adding that the administration would continue its "zero tolerance policy" of criminally prosecuting anyone who crosses the border illegally.
He said he had been swayed by images of children who have been taken from parents jailed and prosecuted for illegal border-crossing.
The executive order calls for:
Vice-President Mike Pence and Department of Homeland Security Secretary Kirstjen Nielsen, who has emerged as the face of the White House policy, were both present.
Ms Nielsen was confronted by protesters at a Mexican restaurant in Washington on Wednesday evening.
In April, the US attorney general announced the policy to criminally charge and jail undocumented border crossers.
As children cannot legally be jailed with their parents, they have been kept in separate facilities.
After initial detention, they were sent to about 100 detention centres in 17 states, most but not all in areas close to the Mexican border.
Three "tender age shelters", all in southern Texas, were set up for babies and toddlers, as well as a tent camp.
Pictures of dozens of children sleeping in fenced enclosures and audio of them crying emerged in recent days, provoking the widespread criticism.
Under previous US administrations, immigrants caught crossing the border for the first time tended to be issued with court summonses and released.
Six US airlines have told the Trump administration not to use their aircraft to transport child migrants separated from their parents.
Alaska, American, Delta, Frontier, Southwest and United all said the policy contravened their values.
On Wednesday New York Mayor Bill de Blasio said he was "shocked" that 239 migrant children had been sent to a single reception centre in Harlem without the knowledge of the city authorities.
Apparently, more children were still on their way to the city. The American Civil Liberties Union, an NGO, tweeted footage of people greeting children arriving at LaGuardia airport.
Mr Trump's order says that families could remain together in detention instead of separating children from parents illegally crossing into the country.
But immigrant children are only allowed to be held for 20 days, according to a 21-year-old landmark court decision known as the Flores agreement.
The order also calls for the justice department to request to modify the Flores agreement to allow children to be held longer, but it is unclear what will happen if the law does not change.
The Trump administration is likely to become tangled in legal battles with immigration activists on behalf of detained migrants if it is unable to overturn the Flores agreement before the 20-day deadline.
Health and Human Services (HHS) Secretary Alex Azar has said his department would begin working to return detained immigrant children to their families, but did not give a timeline.
A top HHS official told Agence France-Presse on Tuesday they had no system in place to do so.