Giuliani, 72, has been one of Trump's most vocal and high-profile supporters, and according to sources close to him and Trump, he is eager to become the top U.S. diplomat and expects a decision by Trump as early as next week. The other top candidate, the sources said, is former United Nations Ambassador John Bolton, a foreign policy hawk.
New York mayor at the time of the Sept. 11, 2001, attacks by al Qaeda militants, Giuliani is also considered a hard-liner on national security matters, but he has little diplomatic experience.
Still, some prominent Republicans said he is qualified to take command of U.S. diplomacy at a time of chaos in the Middle East, rising nationalism in much of Europe, and growing challenges from Russia and China.
Senator Lindsey Graham, a respected conservative voice on defense and foreign policy who serves on the Senate Armed Services Committee and who made a rather short-lived bid for the 2016 Republican presidential nomination, on Tuesday called Giuliani "competent and capable" of being secretary of state.
"Rudy is an internationally-known figure. He's a personal friend. He has dealt with the unimaginable, which was 9/11. He's a loyal supporter of President Trump. He should be rewarded in my view," Graham told reporters.
Giuliani himself extolled his foreign policy credentials in an interview with Fox News Channel's Sean Hannity on Tuesday.
"I've been in 80 countries, 150 different foreign trips," Giuliani said. "A lot of it for different reasons. Speeches. Security consulting, where I helped bring down crime."
Critics, however, said they are troubled not only by Giuliani's combative nature and lack of experience, but also by his international business ties and his lucrative speaking engagements for an Iranian exile group that was on the U.S. terrorism list until four years ago.
The Trump transition team did not immediately respond to a request for comment on its consideration of Giuliani for secretary of state.
After serving as New York mayor for eight years, Giuliani founded management and security consulting firm Giuliani Partners in 2002, which he left in 2007 when he campaigned for the Republican presidential nomination and questions were raised about his foreign business ties.
The firm's clients have included Colombia and, reportedly, the government of Qatar. Giuliani appears to have resumed work with the firm after his 2007 failed presidential bid, and is listed as chairman and chief executive officer of Giuliani Partners on the Giuliani Security and Safety website.
He joined a Texas law firm as a name partner in 2005. The firm did lobbying work for Citgo, a U.S. subsidiary of Venezuela's state-owned oil company, which at the time was controlled by President Hugo Chavez, Venezuela's late socialist ruler. At the time, Giuliani's office said he was not personally involved in the lobbying and said Giuliani believed that "Chavez is not a friend of the United States."
Giuliani's dealings in Russia may face scrutiny in Senate confirmation hearings. His ties to TriGlobal Strategic Ventures, a consulting firm that helps Western clients advance their business interests in emerging markets of the former Soviet Union, date back to 2004, when Giuliani visited Moscow to meet Russian businessmen and politicians, according to the company's website.
The consulting firm's president, Vitaly Pruss, has "created and developed strategies" for companies including Russian oil pipeline monopoly Transneft and has "worked closely" with Giuliani Partners, according to his profile on TriGlobal's website. State-owned Transneft was among Russian oil companies targeted with sanctions by Western powers following Russia's annexation of Crimea under President Vladimir Putin.
Giuliani also has spoken in support of the People's Mujahedin Organization of Iran, a group of Islamic leftists who opposed Iran's late shah, but fell out with the Shi'ite clerics who took power after the 1979 revolution and later aligned itself with Iraqi leader Saddam Hussein. The U.S. government considered it a terrorist organization until 2012.
Senator Rand Paul of Kentucky, who also was a contender for the 2016 Republican presidential nomination, told CNN on Tuesday he was worried about Giuliani's ties to foreign governments.
"Whether or not you have divided loyalties obviously is very important," Paul said. "I hope Donald Trump will pick somebody consistent with what he said on the campaign trail - Iraq war was a mistake, regime change in the Middle East is a mistake.
"You want to have a diplomat in charge of diplomacy," he said.
Critics, however, say the bottom line is that Giuliani is no diplomat, either personally or professionally.