The proposed rule could be published in the Federal Register as early as Monday, followed by a 30-day comment period. After that, the comments will close and it will become final rule .

Critics, such as Ms. Brogan-Kator, said the rule would allow organizations to place their personal religious beliefs above the needs of children in their care, but the administration countered that it was not preventing L.G.B.T. people from adopting.

“The administration is rolling back an Obama-era rule that was proposed in the 12 o’clock hour of the last administration that jeopardizes the ability of faith-based providers to continue serving their communities,” the White House said in a statement on Saturday. “The federal government should not be in the business of forcing child welfare providers to choose between helping children and their faith.”

According to the Adoption Network, there are more than 400,000 children in the foster care system in the United States. More than 114,000 cannot be returned to their families and are waiting to be adopted.

The Williams Institute at UCLA School of Law estimated in a report that 114,000 same-sex couples in 2016 were raising children in the United States. Same-sex couples with children were far more likely than different-sex couples with children to have an adopted child, 21.4 percent versus 3 percent, the report found.