"I think the Iraq situation...is maybe the most underestimated of all the things we're looking at," Frederick Kempe, president and CEO of the Atlantic Council, said on CNBC's "Capital Connection" on Monday.
"There's actually an opening for Iraq to create more independence for itself, from Iran," he said. "That's what I'd be watching."
His comments came amid heightened tensions in the Middle East following an American airstrike in Iraq that killed Iran's top commander Qasem Soleimani. Tehran retaliated by attacking U.S. targets in Iraq, but both sides now appear to have backed away from military actions.
"If you listen to some members of the U.S. government, they believe that Iraq over time has fallen more and more under the sway of Iran," he said. America has been trying to push Iraq toward being more energy independent, but faced opposition from two individuals, he added.
"One of them was named Soleimani, one of them was named Muhandis," he said, referring to Iraqi militia leader Abu Mahdi al-Muhandis who was killed in the same airstrike. "They're both gone."
"(The infrastructure and militia forces are) still all there, but the leadership that was so crucial is gone," he said.
The prospect of Iraq shaking off Iranian influence would decrease, however, if U.S. forces leave, he predicted. Iraq's prime minister last week asked the United States to start working on withdrawing troops from the country.
Kempe said that isn't want President Donald Trump would want, and would not happen "overnight" or "at all."
"I do think that it would be unfortunate because that, of course, would open the way for Iran to take even more control, at a point where the deaths of these two individuals could create more independence for Iraq from Iran."
Kempe also said it was a "lucky break" that Soleimani was "off the stage" when Oman's Sultan Qaboos bin Said died.
"This is the kind of situation Soleimani would have taken advantage of," he said. "Trying to shape the future leadership, maybe even insinuating themselves so that Iran could have a veto or have a real impact in the future of Oman."
He added: "The fact that Soleimani was off the stage at this point was…a chance of history. But I think Oman has come out better for this as well."
— AP and Reuters contributed to this report.