The position would give Moore a vote at the policy-setting table of an institution whose interest rate hikes previous year were a frequent target of Trump's ire.
The Fed's key benchmark rate is now 2.25 to 2.5 percent.
Moore is a visiting fellow for Project for Economic Growth at The Heritage Foundation.
Asked whether Trump might want him on the Fed's board to check Powell's influence and protect the administration's economic policies, Moore said, "The only thing (Trump) told me was, pursue policies that are good for the American workers".
"I believe the people on the Federal Reserve Board should be thrown out for economic malpractice", he said on radio in December. The Fed chairman at the time, Ben Bernanke, and Janet Yellen, who served as Bernanke's deputy before a four-year stint as chair, saw the dot plot as a way of giving markets a look into the Fed's thinking beyond any immediate decision-making.
In a TV interview, he called the Fed's decision to raise rates in December a "substantial mistake" while saying that he did not want to be a disruptor and "I want to be somebody who can really help chairman Powell".
Moore, like Trump, has been a frequent critic of Fed Chairman Jerome Powell.
As a Fed governor, a position that requires Senate confirmation, Moore would be in a position to espouse his and the White Houses view that the economy can grow much faster without generating inflation.
As a TV commentator, Moore has been an anti-tax conservative and fiscal hawk. And he contended that the approach, if adopted, would help accelerate economic growth above 3 percent, compared with the longer-run average of 1.9 percent that Fed officials have forecast.
Fed officials on Wednesday penciled in a long-term growth estimate of 1.9 per cent for the United States economy.
"I will be nominating Mr. Moore to the Fed", Trump told reporters as he arrived in Florida for a weekend trip.
But if other economic data, including from the labor market, along with inflation readings "suggest that the economy may be overheating, then I think that I'd be comfortable to move", he said.
The Fed has become increasingly anxious about meeting its 2 percent inflation target and skeptical that the Trump administration's tax cuts and deregulation will unleash faster economic growth.
The US Fed on Wednesday left interest rates unchanged after concluding a two-day policy meeting, and announced that it will end the runoff of its balance sheet at the end of September.
The spread between yields on three-month Treasury bills and 10-year notes fell below zero for the first time since 2007 after USA manufacturing data missed estimates.
Moore joined CNN as a contributor in early 2017.