“Well, I have a solution,” Mr. Trump replied. “Tell people not to come to our country illegally. That’s the solution. Don’t come to our country illegally. Come like other people do. Come legally.”

A reporter asked him if he was suggesting that the children be punished.

“I’m saying this, very simply: We have laws,” Mr. Trump said. “We have borders. Don’t come to our country illegally. It’s not a good thing. And as far as ICE is concerned, the people that are fighting ICE, it’s a disgrace. These people go into harm’s way. There is nobody under greater danger than the people from ICE.”

The children involved in Tuesday’s reunions are some of the youngest immigrants caught up in the Trump administration’s family separation policy. They had been apprehended with their relatives at the border, separated from their families and detained in so-called tender-age shelters for children under the age of 12. They have been separated from their loved ones, in many cases, for weeks.

The Texas-based nonprofit that has raised more than $20 million to help families separated at the border said Tuesday that it plans to offer the money as bond payments to release thousands of mothers who remain in custody.

Leaders of the group, Raices, said they planned to offer to pay as much of the money as was necessary to reunite an estimated 2,500 children and mothers. They have already paid nearly $100,000 in bonds for roughly 18 people who are in custody with the Department of Homeland Security.

Officials with Raices acknowledge that they may face resistance from the Department of Homeland Security in accepting bond payments. But they hope to use the issue to keep attention on the children who remain scattered across the country, and to press Congress to take action.

“We will not simply fork over a check and hope for the best,” said Jonathan Ryan, the executive director of Raices. “We expect that the government will act in good faith, as the American people are demanding, and sit down with us, put all of our cards on the table.”