The president last month offered a Fed position to another ally, conservative economist Stephen Moore.
Cain, a former CEO of Godfather's Pizza, ran for the 2012 Republican presidential nomination but dropped out after allegations of sexual harassment and infidelity.
One of the officials told Axios that Trump "won't formally announce until the vet" of Cain "is completed".
Mr Trump has been accused of putting forward political loyalists to the Fed.
There are now two vacancies on the seven-member Fed Board.
Cain has recently been running a pro-Trump political action committee, America Fighting Back; Moore was a former Trump economic adviser during his 2016 campaign.
Former Kansas City Fed President Thomas Hoenig described the role of fed directors like Cain as representatives of "Main Street business, community development, organized labor and financial services sectors" who help the Federal Reserve better understand the economy and oversee operations. It features a photo of the president on its website and says, "We must protect Donald Trump and his agenda from impeachment". He advocated for the U.S.to return to the gold standard during his presidential campaign and as recently as December 2017 defended higher interest rates, a position that contrasts with Trump's repeated criticisms of the Fed a year ago.
On Wednesday, White House economic advisor Larry Kudlow reinforced the president' support for Moore, despite a New York Times report that he failed to fully pay taxes and alimony. Doing so, however, raises the possibility of Capitol Hill testimonies regarding sexual harassment allegations against Cain, which ultimately derailed his 2012 candidacy for the Republican presidential nomination.
During his 2012 presidential campaign, Cain famously touted an economic plan, the "9-9-9 Plan", which would have replaced the USA tax system with a flat 9 percent business, income and national sales tax. I find Herman to be an outstanding person.
Trump has repeatedly denounced the Fed leadership of Powell, whom he selected as chairman after deciding not to re-nominate Janet Yellen. Asked if he was sending any signal to the Fed by putting up Mr Cain and Mr Moore, the President said "none whatsoever".
Trump's other nominees have held views more in line with traditional selections for the Fed board. The White House announced Thursday that Trump was nominating Michelle Bowman, a Kansas banking regulator, for a full 14-year term on the Fed. "But if it were a nomination, you can bet [what] the interest rates he would be pushing for.If Herman Cain were on the Fed, you'd know the interest rate would soon be 9-9-9". She joined the Fed last year, taking over a term that will end next year. Trump's other Fed appointees have also supported the Powell Fed's rate hikes, which the president has said hurt the economy.
Expectations are mounting among investors that the Fed could even cut its benchmark lending rates.