But a court fight in Britain has delayed their transfer. The lawsuit is over whether the British government may share evidence with the United States without an assurance that American prosecutors will not seek the death penalty.

The British government has shown witness statements about the two men to the Justice Department, but testimony from British government officials would also probably be necessary at any trial. Mr. Elsheikh’s mother has filed a lawsuit seeking to block such cooperation because the United States government has not promised it will not seek to execute her son. Britain has abolished the death penalty.

The American military had been making contingency plans to get a list of about five dozen of the highest-priority detainees from that group out of northern Syria since December, when Mr. Trump first announced that he would withdraw troops from the country before his administration slowed down that plan, one official said.

Mr. Trump’s decision to let Turkey proceed prompted the military to start getting those prisoners out, lest they escape amid the chaos and as the Kurds pull guards out of the prisons to help fight. But the detainees were scattered among numerous makeshift prisons, and it was not clear how many on the list would ultimately be taken, the official said.

The Washington Post earlier reported on the move to transfer custody of detainees, including the two British men.

Mr. Trump’s decision to clear the way for Turkey to launch its operation into northern Syria is bringing to an abrupt crisis a long-simmering problem: About 50 countries have citizens in the Kurds’ prisons for ISIS fighters — and in the displaced persons camps where tens of thousands of ISIS women and children are held — and have been reluctant to repatriate them, instead leaving them in the Kurds’ hands indefinitely.

The male fighters the Kurds are holding include about 9,000 local Syrians and Iraqis, as well as 2,000 foreign fighters — including scores from Western Europe. Many European law enforcement officials fear that if they repatriate their extremist citizens, they would be unable to convict them or keep them locked up for a long time.