“There will not be a grandfathering of existing cases,” said Kenneth Wolfe, a spokesman for the Administration for Children and Families, a division of the Department of Health and Human Services. Mr. Wolfe said the decision about the children was made by the White House.

But later Wednesday evening, Brian Marriott, the senior director of communications for the agency, said that Mr. Wolfe had “misspoke” and insisted that “it is still very early, and we are awaiting further guidance on the matter.” Mr. Marriott said that “reunification is always the goal” and that the agency “is working toward that” for the children separated from their families because of Mr. Trump’s policy.

His statement left open the possibility, though, that the children could be connected with other family members or “appropriate” sponsors living in the United States, not necessarily the parent they were separated from at the border.

The president signed the executive order days after he said that the only way to end the division of families was through congressional action because “you can’t do it through an executive order.” But he changed his mind after a barrage of criticism from Democrats, activists, members of his own party and even his wife and eldest daughter, who privately told him the policy was wrong.

Mr. Trump had previously been told by advisers that there was no way for the policy to be changed through an executive order, and it was unclear what the genesis of the measure he signed was. But the White House counsel, Donald F. McGahn II, had concerns about moving ahead with an executive order that would face an uphill battle in the courts.