"General James "Mad Dog" Mattis, who is being considered for Secretary of Defence, was very impressive yesterday. A true General's General!" Trump said in a tweet.
Mattis, former Central Command leader, was one of several people who have met Trump and Vice president-elect Mike Pence.
The 66-year-old Marine Corps general, who memorably said "it's fun to shoot some people", would add to a cabinet of national security super-hawks and signal a return to a more aggressive defence of American interests abroad.
In Mattis, Trump has a candidate who was held in high regard throughout the ranks of the Marine Corps during his 44 years of service. A seasoned combat commander, he led a task force into southern Afghanistan in 2001 and a Marine division at the time of the Iraq invasion in 2003, according to CNN.
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Mattis would require a waiver from Congress to be eligible for the post, since a retired officer is needed to be out of uniform for at least seven years before taking control of the Pentagon responsibilities.
With Republicans in control on Capitol Hill and the general praise for the general's career, the waiver would likely be a formality to obtain.
Appearing on Fox News, Pence said Romney, 69, is being considered for the position of Secretary of State.
Governor "Mitt Romney came in. They had a good meeting. It was a warm and substantive exchange. I know he's under active consideration to be the secretary of state of the United States along with other distinguished Americans," Pence said.
"We spent about an hour together with the team, the president-elect and I on one side of the table and some of the team on the other side with Governor Romney. And we talked through a lot of substantive issues," he said.
"From there, we'll work on issues ranging from ending illegal immigration, reviving our economy through tax reform, rebuilding the military, restoring the infrastructure of this country," the vice president-elect said.