The source said John Bolton, who served as U.S. ambassador to the United Nations under President George W. Bush, was also under consideration to head the State Department.
Giuliani, 72, was one of Trump’s most vocal and high-profile supporters on the campaign trail, introducing him frequently at rallies with slashing attacks against Democrat Hillary Clinton.
He was New York’s mayor during the Sept. 11, 2001, attacks and led the city during the crisis after the destruction of the World Trade Center twin towers, working closely with then-President George W. Bush.
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Giuliani was an early supporter of Trump, has been his friend for many years, and defended him vigorously through the various controversies that ensnared the New York businessman.
His rhetoric was notable for repeatedly accusing Clinton of having violated laws through her handling of classified information on a private email server while she was secretary of state from 2009 to 2013.
“If Rudy wants it, he’ll get it,” former House of Representatives Speaker Newt Gingrich said on Fox News.