At the White House, senior officials have expressed concern that the latest allegations could set back relations with Germany just as Mr. Obama and Ms. Merkel are struggling to move past the distrust generated by the Snowden disclosures, including the revelation that the N.S.A. had tapped Ms. Merkel’s cellphone.

What is particularly baffling to these officials is that the C.I.A. did not inform the White House that its agent — a 31-year-old employee of Germany’s federal intelligence service, the BND — had been compromised, given his arrest the day before the two leaders spoke. According to German news media reports, the agency may have been aware three weeks before the arrest that the German authorities were monitoring the man.

A central question, one American official said, is how high the information about the agent went in the C.I.A.’s command — whether it was bottled up at the level of the station chief in Berlin or transmitted to senior officials, including the director, John O. Brennan, who is responsible for briefing the White House.