Former New York City Mayor Rudy Giuliani was the favorite to be secretary of state in President-elect Donald Trump's administration, a senior Trump transition official said Monday.

The official told the Associated Press there was no real competition for the job and that it was Giuliani's if he wanted it. However, a second official cautioned that John Bolton, a former U.S. ambassador to the United Nations, remained in contention for the job.

A senior source told Fox News that Giuliani was being considered for the secretary of state job, but said the choice was not locked in. The source added that Sen. Jeff Sessions, R-Ala., one of Trump's earliest Washington supporters, was getting a lot of say in the selection of officeholders.

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Giuliani, a top Trump surrogate, said he "won't be attorney general" in a Trump administration at a Washington event sponsored by the Wall Street Journal. The former federal prosecutor had been seen as a top contender for the Justice Department post before Trump's election last week.

Giuliani said he thought Bolton "would be a very good choice" for secretary of state. But asked if there was anyone better, he replied with a mischievous smile: "Maybe me, I don't know."

During the event, Giuliani said that defeating the ISIS terror group was Trump's top foreign policy priority, though he did not go into specifics of the president-elect's plan. The former mayor also discussed Russia's global power and influence.

"Russia thinks it’s a military competitor, it really isn’t," Giuliani said. "It’s our unwillingness under Obama to even threaten the use of our military that makes Russia so powerful."

Giuliani, 72, would be an out-of-box choice to lead the State Department due to his lack of extensive foreign policy experience. Known for his hard-line law-and-order views and brusque manner, he would set a very different tone than previous holders of the job, including Trump's ex-rival Hillary Clinton, Colin Powell and Condoleezza Rice.

Bolton has years of federal government experience, but he has also raised eyebrows with some of his hawkish stances, including a 2015 op-ed in The New York Times in which he advocated bombing Iran to halt the country's development of nuclear weapons.

Trump was also considering Monday whether to inject new diversity into the GOP by recommending a woman to lead the Republican Party and an openly gay man to represent the United States at the United Nations.

The moves, among dozens under consideration from his transition team, follow an intense and extended backlash from Trump's decision on Sunday to appoint Steve Bannon, a man celebrated by the white nationalist movement, to serve as his chief strategist and senior adviser.

"After winning the presidency but losing the popular vote, President-elect Trump must try to bring Americans together — not continue to fan the flames of division and bigotry," said House Democratic leader Nancy Pelosi. She called Bannon's appointment "an alarming signal" that Trump "remains committed to the hateful and divisive vision that defined his campaign."

His inauguration just 66 days away, however, Trump focused on building his team and speaking to foreign leaders. He remained sequestered in Trump Tower in New York.

Inexperienced on the international stage, the Republican president-elect spoke to Russian President Vladimir Putin on the phone. His transition office said in a readout that "he is very much looking forward to having a strong and enduring relationship with Russia and the people of Russia." Trump has spoken in recent days with the leaders of China, Mexico, South Korea and Canada.

At the same time, Trump was considering tapping Richard Grenell as U.S. ambassador to the United Nations. He would be the first openly gay person to fill a Cabinet-level foreign policy post. Grenell, known in part for aggressive criticism of rivals on Twitter, previously served as U.S. spokesman at the U.N. under President George W. Bush.

Trump was also weighing whether to select Michigan GOP chairwoman Ronna Romney McDaniel, a niece of chief Trump critic and 2012 presidential nominee Mitt Romney. She would be the second woman ever to lead the Republican National Committee — and the first in four decades.

"I'll be interested in whatever Mr. Trump wants," McDaniel told The Associated Press on Monday, adding that she was planning to seek the Michigan GOP chairmanship again.

Appointing McDaniel to run the GOP's political arm could be an effort to help the party heal the anger after a campaign in which Trump demeaned women. The appointment of Grenell, who has openly supported same-sex marriage, could begin to ease concerns by the gay community about Vice President-elect Mike Pence's opposition to same-sex marriage during his time as Indiana governor.

The personnel moves under consideration were confirmed by people with direct knowledge of Trump's thinking who were not authorized to publicly disclose private discussions. They stressed that the decisions were not final.

Fox News' Serafin Gomez and the Associated Press contributed to this report.

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