Petraeus, one of America's most celebrated generals credited with turning the tide in Iraq, spent an hour with Trump at his office in Trump Tower in Manhattan yesterday.
Trump tweeted about the meeting immediately afterwards, saying he "was very impressed" with the retired general.
"Just met with General Petraeus--was very impressed!" Trump said in a tweet after his meeting with the former decorated general of the US armed forces and the top spy master.
He said Trump showed a great grasp of a variety of global challenges.
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"I was with him for about an hour. He basically walked us around the world," he said.
"(Trump) showed a great grasp of a variety of the challenges that are out there and some of the opportunities as well. Very good conversation and we'll see where it goes from here. We'll see where it goes from here," Petraeus said amidst reports that the former CIA director is among the new names for his Secretary of State.
Trump is scheduled to have another round of meeting with his former bitter opponent and 2012 presidential candidate Mitt Romney.
"The search process or the interview process as the president-elect works to come to a final decision on the secretary of state of front, he'll complete that and announce his decision when he's ready," the transition communication director Jason Miller told reporters during a conference call.
"He has met with a number of very well-qualified individuals and obviously through this week met with a number of well-qualified individuals and he's going to pick the person who he thinks will do the absolute best job," Miller said.
According to reports, Petraeus's appointment could allow for a compromise amid reports of divisions within the Trump team, with some backing Romney and others supporting Rudy Giuliani.
Petraeus agreed to a plea deal that saw him sentenced to
two years' probation and forced him to pay a USD 100,000 fine for sharing classified information.
Before becoming the CIA Director, Petraeus was commander of the US Central Command and also headed the US and NATO troops in Afghanistan and Iraq.
"Indeed, Petraeus would be an outstanding first diplomat of the United States, although Trump is reportedly also considering candidates such as former Republican presidential candidate Mitt Romney, among others," Bergan wrote.