An official said there was no real competition for the job and that it was Mr Giuliani’s if he wanted it.
However, a second official cautioned that John Bolton, a former US ambassador to the United Nations, remains in contention for the position.
Mr Giuliani, 72, would be a left-field choice to lead the US State Department.
A former mayor, federal prosecutor and top Trump adviser, he lacks extensive foreign policy experience. Known for his hard-line law-and-order views and brusque manner, he would set a very different tone to previous holders of the job, including Mr Trump’s ex-rival Hillary Clinton, Colin Powell and Condoleezza Rice.
Mr Bolton has years of federal government experience, but he has also raised eyebrows with some of his hawkish stances, including a 2015 column in The New York Times in which he advocated bombing Iran to halt the country’s development of nuclear weapons.
Mr Giuliani said yesterday at a gathering of CEOs sponsored by the Wall Street Journal that he “won’t be attorney general” in Mr Trump’s administration - a job for which he had long been seen as a top contender.
Asked about the Secretary of State speculation, Mr Giuliani said that Mr Bolton “would be a very good choice”. But when asked if there was anyone better, Mr Giuliani replied with a mischievous smile: “Maybe me, I don’t know.”
Mr Trump is also considering whether to inject new diversity into his party by recommending a woman to head its National Committee, and an openly gay man to represent America at the United Nations.
The moves follow an intense backlash after Mr Trump’s decision to appoint Steve Bannon - a man celebrated by the white nationalist movement - to serve as his chief strategist and senior adviser.
House Democratic leader Nancy Pelosi said: “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.”
She called Mr Bannon’s appointment “an alarming signal” that Donald Trump “remains committed to the hateful and divisive vision that defined his campaign”.
With his inauguration just 66 days away, Mr Trump has focused on building his team and speaking to foreign leaders from Trump Tower in New York.
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The Republican president-elect spoke to Russian leader Vladimir Putin on the phone. His transition office said that “he is very much looking forward to having a strong and enduring relationship with Russia and the people of Russia”.
Mr Trump has spoken in recent days with the leaders of China, Mexico, South Korea and Canada.
At the same time, Mr Trump is considering Richard Grenell for the role of US ambassador to the United Nations, which would make him the first openly gay person to fill a Cabinet-level foreign policy post. Mr Grenell, known in part for aggressive criticism of rivals on Twitter, previously served as US spokesman at the UN under former president George W Bush.
Mr Trump is also weighing whether to select Michigan Republican party chairwoman Ronna Romney McDaniel, a niece of chief Trump critic and 2012 presidential nominee Mitt Romney, to lead the Republican National Committee (RNC). If appointed, she would be the second woman ever in the job - and the first in four decades.
“I’ll be interested in whatever Mr Trump wants,” Ms McDaniel said.
Appointing her to run the RNC could be part of an effort to help the party heal after a bruising election campaign in which Mr Trump came under fire for demeaning women. The appointment of Mr Grenell, who has openly supported same-sex marriage, could ease concerns among the gay community about vice president-elect Mike Pence’s opposition to same-sex marriage during his time as Indiana governor.
The decisions about staffing come a day after Mr Trump made overtures to warring Republican circles by appointing Mr Bannon and RNC chairman Reince Priebus as his White House chief of staff.
The former media executive led a website that appealed to the so-called “alt-right” - a movement often associated with efforts on the far right to preserve “white identity,” oppose multiculturalism, and defend “Western values”.
Mr Priebus defended the media mogul, saying the two made an effective pair as they steered Mr Trump past his Democratic rival Hillary Clinton and towards the presidency. He sought to distance Mr Bannon from the incendiary headlines on his website, saying they were written by unspecified others.
Mr Priebus said: “Together, we’ve been able to manage a lot of the decision making in regard to the campaign. It’s worked very, very well.”
President Barack Obama avoided any direct criticism of Mr Trump’s personnel moves during an afternoon news conference, suggesting that the new president deserves “room to staff up”.
“It’s important for us to let him make his decisions,” said Mr Obama.
“The American people will judge over the course of the next couple of years whether they like what they see.”