Nearly every aspiring ambassador contacted for this article did not return phone calls or declined to comment about any interest in specific jobs. But speculation about who is in line for what often makes its way into the press; last month, The Hollywood Reporter published the names of several West Coast donors said to be on Mr. Obama’s short list for diplomatic posts — a list as closely scrutinized by Hollywood for who wasn’t on it, other donors said, as for who was.

For some would-be diplomats, the hunt began the day after Mr. Obama’s re-election in November, when the president’s top aides began asking his leading fund-raisers if they had interest in serving. Mr. Barzun has been an informal facilitator for donors and fund-raisers, along with Rufus Gifford, his counterpart on Mr. Obama’s campaign staff. Inside the White House, a small group of senior aides — including the counselor Pete Rouse; Valerie Jarrett, a senior adviser; the deputy chief of staff, Alyssa Mastromonaco; Brian McKeon, a national security staff member; and Mr. Obama’s personnel director, Nancy Hogan — have met regularly to discuss potential choices.

Mr. Obama has followed recent tradition in making appointments; like every president going back to Ronald Reagan, he has filled about 70 percent of the posts with career diplomats and 30 percent with political appointees, often but not always top donors. Dangerous spots like Yemen are invariably filled by diplomats, according to statistics compiled by the American Foreign Service Association. Highly sought European and Caribbean countries usually go to political appointees. At least three Obama fund-raisers are interested in Italy this time around, according to people familiar with the roster of potential candidates. They include Azita Raji, a San Francisco philanthropist; John R. Phillips, a Washington lawyer married to the former Obama aide Linda Douglass; and Robert Mailer Anderson, a novelist.

But Asian countries are increasingly desirable. Steve Westly, a California venture capitalist and top Obama fund-raiser, has discussed with administration officials the ambassadorship to China, among other jobs, according to people with knowledge of the talks. The post is currently filled by Gary Locke, the former commerce secretary, who may depart later this year.