He said the pope’s friendly meeting with Ms. Davis suggested “that this frame does not fit.”

Liberal Catholic commentators were left asking whether the pope had been trying to make a statement about religious liberty or same-sex marriage by meeting with Ms. Davis, and if so, why the meeting had been kept secret. Some called it a mistake.

“The news that Pope Francis met privately in Washington, D.C., with Kim Davis throws a wet blanket on the good will that the pontiff had garnered during his U.S. visit last week,” said Francis DeBernardo, executive director of New Ways Ministry, an advocacy group for gay Catholics.

The pope’s meeting with Ms. Davis appears to have been carefully planned. Her lawyer, Mr. Staver, said the idea was first discussed with representatives of the pope on Sept. 14, more than a week before Francis’ arrival in the United States. Mr. Staver declined to say who had proposed it, but said that Vatican officials had been aware of Ms. Davis’s jailing and that the meeting had been arranged through them — not through American bishops or the bishops’ conference in the United States.

“I would not have imagined that this would be a possibility, that the pope, in his incredibly busy schedule here in the States, would have time to meet with Kim Davis,” Mr. Staver said.

Ms. Davis and her husband were scheduled to be in Washington on Friday to pick up an award from the conservative Family Research Council, but flew in a day earlier to meet with Francis, Mr. Staver said. He said he drove the couple to a rendezvous spot in or near the Vatican Embassy, which was surrounded by security and crowds of people hoping for a glimpse of Francis. Sneaking her inside was no mean feat, Mr. Staver said, because Ms. Davis is now nationally recognized.

While waiting in the embassy, the Davises took cellphone pictures of themselves with a portrait of Francis. But, Mr. Staver said, “out of deference and respect, they didn’t want to pull out a cellphone with the pope.”