REUTERS

Mr. Trump's aides have also considered former U.S. Ambassador to the United Nations John Bolton as a possible candidate, but the close relationship between Messrs. Giuliani and Trump was a major consideration, the people said.

Asked at The Wall Street Journal CEO Council meeting in Washington on Monday evening if his title would soon be "Secretary," Mr. Giuliani responded, "One never knows."

Read alsoNYT: Trump picks Priebus as chief of staff and Bannon as strategistMr. Giuliani also said that Mr. Bolton would be a good choice for secretary of state. He was then asked if there was a better choice than Mr. Bolton and he replied, "Maybe me, I don't know."

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For Mr. Trump, it is a choice between a longtime friend and ally in New York, Mr. Giuliani, and a hawkish conservative diplomat, Mr. Bolton, who called last year for the U.S. to bomb Iran. A final decision could be several weeks away, these people said, and other candidates could still emerge.