International condemnation is growing against the US policy of separating migrant families at its border.
UK Prime Minister Theresa May said holding children "in what appear to be cages [is] deeply disturbing" and "wrong".
Pope Francis said he supported US Catholic bishops who said it was "immoral".
The Republican-controlled Congress is under pressure to change President Donald Trump's policy.
Republicans are scrambling to draft a new immigration bill to address the growing outcry.
President Trump told a group of House Republicans he would back any bill they passed, a White House spokesman said.
The president made clear that separating parents and children was "certainly not an attractive thing and does look bad", Representative Tom Cole said.
However, Mr Trump insists it is necessary to stop illegal border crossings.
On Tuesday, he said: "When you prosecute the parents for coming in illegally - which should happen - you have to take the children away,"
Why the outcry?
Pictures of dozens of children sleeping in cages and audio of children crying for their parents have emerged in recent days, provoking the widespread criticism.
It comes after a "zero-tolerance" crackdown on illegal immigration was brought in by Attorney General Jeff Sessions.
US immigration officials say 2,342 children were separated from 2,206 parents between 5 May and 9 June.
Babies and toddlers have been sent to three "tender age" shelters after being separated from their parents, the Associated Press news agency reports.
Doctors and lawyers who visited the shelters described the infants as hysterical and crying, according to the AP report.
Who else has spoken out?
On Tuesday Mexican Foreign Minister Luis Videgaray Caso said the practice was "cruel and inhuman", and clearly violated human rights.
The US Chamber of Commerce and the Business Roundtable, two leading business groups, said the policy was "contrary to American values".
French far-right leader Marine Le Pen, who in the past has praised Mr Trump's policy of curbing immigration, told French TV that she disagreed with splitting children from parents.
A new Reuters/Ipsos opinion poll suggests that most Americans oppose the policy, with 28% supporting it.
What is the policy?
Under the "zero-tolerance" approach unveiled in May, all undocumented border crossers are criminally charged and jailed.
Migrant children cannot be held with their parents, and are kept in separate facilities maintained by the Department of Health and Human Services.
Under previous US administrations, undocumented immigrants caught crossing the border for the first time tended to be issued with court summonses.
But the Trump administration points out that most of those migrants never showed up for court.
The White House has been disputing the terminology used to describe its crackdown.
It describes the holding centres where children have been pictured in metal chain-link enclosures with concrete floors as "shelters" instead of "cages".
What are Trump and lawmakers doing?
Mr Trump will meet Congress members on Wednesday afternoon to discuss a more moderate bill.
The House of Representatives is expecting to vote on it later this week.
The compromise would limit, but not ban outright, family separations. It would also offer an eventual path to citizenship for undocumented adult immigrants, known as Dreamers, who entered the US as children.
The Republican legislation would also provide $25bn (£19bn) in funding for border security, including Mr Trump's planned US-Mexico wall.
House Republicans were reportedly working on a revised plan that would mean children would be detained for longer than is currently allowed but kept with their parents, the Associated Press reports.
Senate Majority Leader Mitch McConnell also said he "hoped to reach out to the Democrats" about crafting a bipartisan deal. However Democrats argue that no congressional action is required and the president can simply reverse his own policy.
Governors from at least eight US states - including two Republicans - have reversed their decision to send National Guard troops to the US-Mexico border.