Under Trump's new "zero tolerance" immigration policy, the children of men and women caught crossing the border illegally are placed in government custody while their parents await prosecution in ICE detention centers.
The uproar has escalated after pictures of children in detention centre cages were released.
Trump also expressed disappointment with America's southern neighbor.
There is no law mandating the separation of children and parents at the border.
Even though he ended an Obama-era program that allowed undocumented immigrants brought as children to avoid deportation, he also said that they were "terrific people" and that he wanted a legislative solution that showed "great heart".
After his remarks, the president raised eyebrows by walking across the stage, arms outstretched, and embracing a U.S. flag.
TRUMP: "The Democrats forced that law upon our nation. They don't care about crime and want illegal immigrants, no matter how bad they may be, to pour into and infest our Country, like MS-13", the president tweeted.
Senator Ted Cruz, a Republican from Texas, announced Monday that he was introducing emergency legislation meant to keep immigrant families together.
One is a hardline measure favored by conservatives, and is unlikely to pass.
As the immigration policy looms large in the national consciousness, border-state lawmakers are feeling the pressure from constituents who are outraged that children are being separated from their parents.
But Trump himself put that bill in doubt.
In this post, we're answering a question we received: Why do they take the children away from their parents at the border?
A group led by Senator Orrin Hatch have written Attorney General Jeff Sessions demanding a pause to the separations, while Senator John Cornyn was drafting what he called "emergency" legislation - backed by leadership - that would allow family units to remain intact during adjudication of their cases.
The United Nations has slammed the separation practice as unconscionable, while Amnesty International blasted it as "nothing short of torture".
A scathing denunciation by the American Academy of Pediatrics called on the departments of Homeland Security and Justice to "immediately end the policy of family separation".
Tuesday's Republican huddle will be closely watched, in part to see whether any lawmakers directly confront the president.
Both Culberson and Hurd are facing competitive races in November.
The president has accused Democrats of provoking the crisis by blocking legislation to combat illegal immigration.
Texan Congresswoman Sheila Lee told CNN she had been to the border and witnessed the "cries and screams of these little ones", as well as mothers saying they feared they would never see their children again.
"Congress should pass legislation to prevent the separation of children from their parents while they are waiting adjudication of their legal status in the USA", he said in a statement.
"Obviously nobody wants to separate parents from children, because it's cruel".
"I don't want judges", he said. Speaking before an event on space Monday, he argued that immigration laws which he blamed for the policy are "horrible and tough". There are now 12,000 children in the custody of the Department of Health and Human Services.
The majority crossed the border without their parents.
Audio obtained by the investigative news outlet ProPublica captures the harrowing cries of 10 Central American children when they were taken from their parents at the USA border by immigration authorities.
But activists seized on the symbolism.
"Politically correct or not, we have a country that needs security, that needs safety, " he said. "I am all alone'".