After each interview, Mr. Trump, dressed in a suit and tie, emerged from the club next to an American flag to see the candidates off to their sport-utility vehicles and once again speak with reporters.

“Tremendous talent — we’re seeing tremendous talent,” Mr. Trump said on Saturday. “People that, as I say, we will ‘make America great again.’ These are really great people. These are really, really talented people.”

By the evening, however, Mr. Trump had announced no appointments, leaving reporters waiting on the cold, gusty day to speculate about Mr. Trump’s brief comments.

“We made a couple of deals,” he said as his weekend of interviews drew to a close.

Mr. Giuliani is apparently in competition with several others for the secretary of state position, including David H. Petraeus, the retired four-star general who served as Mr. Obama’s C.I.A. director before leaving amid revelations that he had provided classified information to a woman with whom he was having an affair.

Mr. Romney, the 2012 Republican presidential nominee and one of Mr. Trump’s fiercest critics during the 2016 campaign, met with the president-elect on Saturday. If he becomes secretary of state, he could be a moderating influence on the hard-liners Mr. Trump has chosen for attorney general, national security adviser and C.I.A. director.

“I can say that Governor Romney is under active and serious consideration to serve as secretary of state of the United States,” Vice President-elect Mike Pence said Sunday on CBS’s “Face the Nation” program.