Larry Kudlow, head of the president's National Economic Council, says the administration is allowed to put people in the bank that discuss the president's perspectives on the market.
Hassett stressed that both Herman Cain and Stephen Moore are now in the "intent to nominate" phase of the process, which means they are subject to vetting and paperwork within the administration ahead of a formal nomination.
After months of rumors, Trump on Thursday confirmed that he is recommending Cain for a Federal Reserve Board seat.
Cain is a controversial choice both for his economic philosophy and for his political baggage, having dropped a White House bid in 2011 amid accusations of sexual misconduct.
"I'm sure the Senate Banking Committee will do likewise", Kudlow said of the allegations, which Cain denies.
Neither man has been formally nominated yet, and have still to undergo background checks.
Comments by Moore saying he'd be on a "steep learning curve" as to how the Fed operates reflected "a little modesty" by the Heritage Foundation official, Kudlow said.
A top economic adviser to President Donald Trump said Friday by appointing a couple of Trump's close political allies to the Fed 36, that the government is not attempting to hurt the liberty of the Federal Reserve. "More than I've ever seen", he responded, appearing to indicate that people make allegations in D.C. against important people, sometimes without reason or cause.
Herman Cain, former chairman and chief executive officer of Godfather's Pizza.
"They have to collect an inordinate amount of information on you, your background, your family, your friends, your animals, your pets, for the last 50 years", he said in the video.
"I think Herman would be a great member of the Fed", Mulvaney said on "Fox News Sunday". "They don't necessarily pan out", he concluded. "I'm not going to litigate that here".
"You better believe that the people who hate me, who do not like conservatism, Republicans, are already digging up all of the negative stuff that's in stories from eight years ago", Cain said.