The cost of the U.S. ban on paying for hostages

“The policy of no concession has always been there, but we used to interpret it in a much more flexible way,” said Gary Noesner, who retired in 2003 as chief of the F.B.I.’s Crisis Negotiation Unit. “The problem in my mind is that we have devalued negotiation as a tool.”

Mr. Abo Aljoud’s account mirrors those of six other witnesses who were either present at the moments of the Americans’ abductions or were held alongside them. They describe going to lengths this spring and summer to give American officials information that they believed could help free the hostages. And they say they were disheartened by what they perceived as a lack of urgency on the part of those officials.

Similarly, several Syrian rebel commanders say that American officials rejected their proposals to act as go-betweens with the jihadists in war-torn Syria, often for a sizable cash fee, on the grounds that the United States does not negotiate with terrorists.

In one instance a few months before Mr. Foley was beheaded in August, a rebel fighter said he had brought an Islamic State sheikh here to Gaziantep, Turkey, about 35 miles north of the Syrian border, where a delegation of American officials was meeting. The sheikh had a letter from the group stating that he was authorized to negotiate — but the officials declined to talk with him.