Trump has been facing an ongoing impeachment inquiry led by House Democrats into what investigators believed to be his inappropriate interactions with Ukraine.
Trump has slammed Cohen as "a weak person" and the Russia probe as a "witch hunt."
Trump will host Moon at a summit at the White House on May 22, and the two are expected to discuss the upcoming US-DPRK summit.
Tria was speaking on the sidelines of the Boao Forum for Asia Rome Conference, attended by 300 political, business and academic leaders across 15 countries. The BRI was a hot topic, with many speakers hailing it as an important stabilizer of globalization trends.
Trump has challenged US intelligence agencies in the past, including casting doubt on their conclusion that Russia interfered in the 2016 US election to help him win the presidency.
Trump's Twitter post appeared not only at odds with the Treasury Department, but with his own national security adviser, John Bolton, a hardliner who DPRK blamed for the breakdown of a second Trump-Kim summit in Hanoi last month.
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Trump and Kim met for the second day at the Hotel Metropole in Hanoi.
Trump has complained frequently that the United States is treated unfairly in global trade and has sharply criticized the WTO for letting that happen.
Trump announced in 2017 that he would begin to pull the US out of the Paris Agreement on combating global warming. Biden says climate change is on his list of major crises facing the US.
Trump is also considering replacing Kelly, a retired Marine Corps general and a supporter of Nielsen, with Vice-President Mike Pence's chief of staff, Nick Ayers, the source said.