Trump -- the 70-year-old Republican billionaire who takes office in just nine weeks -- was to meet with Vice President-elect Mike Pence later today in New York to discuss the next round of cabinet appointments.
Giuliani, a member of Trump's inner circle, had been considered a leading candidate for attorney general, but at a public forum in Washington yesterday, he said he would not be heading the Justice Department.
The 72-year-old Giuliani was mayor of New York on 9/11, and his decisive leadership after the World Trade Center's twin towers were toppled in the September 2001 attacks made him a national hero.
A crime-fighting former prosecutor who cleaned up New York and lifted its fortunes, Giuliani made a bid for the Republican presidential nomination in 2008 but withdrew after learning he had prostate cancer.
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"John would be a very good choice," Giuliani said at the forum sponsored by The Wall Street Journal.
Asked if there were anybody better, Giuliani quipped: "Maybe me, I don't know."
Bolton made no mention of his chances in an interview today with Fox News but seemed like he was auditioning for it, weighing in heavily on US relations with Moscow the day after Trump called Russian President Vladimir Putin.
"I think one of the reasons that Putin is very casual about expanding Russian influence, taking advantage of America in Eastern Europe and the Middle East is that he sees, quite correctly, Barack Obama as an exceedingly weak leader," he said.
Bolton was a controversial choice for UN envoy in 2005, having once said if the UN headquarters lost 10 floors, "it wouldn't make a bit of difference."
Also in the mix for a top job is Senator Jeff Sessions of Alabama, an early Trump supporter who is reported to be under consideration possibly for attorney general, secretary of defense or head of the Department of Homeland Security.
Sessions has been a fierce advocate for restrictions on immigration, but was once rejected for a federal judgeship after officials testified he made racist remarks, The New York Times reported.