That assurance followed Hassett's appearance on Fox Business, where the head of Trump's Council of Economic Advisers said he is highly confident that the president is happy with Mnuchin.
Those remarks came amid the stock market's suffering a significant decline in prices.
As markets struggled to gain a foothold Wednesday morning, the partial shutdown of the federal government entered its fifth day, with no immediate end in sight.
On Tuesday, Trump praised US companies and said their lower stock prices present an opportunity for investors.
Trump, however, said on Christmas Day that he had confidence in Mnuchin, telling reporters that Mnuchin is a "very talented, very smart person".
When asked by local news media about whether Powell's job as Fed chairman was safe, Hassett responded: "Yes, of course".
Mr Trump blasted the Fed again, this time saying it was the United States economy's "only problem", and investors have closely followed reports the president had privately discussed the possibility of firing Mr Powell.
Trump doesn't want the public to think the stock market has tanked because of Trump's government shutdown, his trade war with China, and the $1.9 trillion increase in the nation's debt caused by his tax cut for corporations and the wealthy.
For the month of December, all of the S&P 500's eleven sectors have been negative.
"The only problem our economy has is the Fed", Trump tweeted a few days ago. Phone calls Mnuchins with the heads of some of the largest American banks seem to have investors, however, rather more anxious. Mnuchin also tweeted that he had checked with the heads of the six largest US banks to ensure they had enough liquidity to operate in a stock market that has tumbled sharply since October. We are absolutely committed to that mission and nothing will deter us from doing what we think is the right thing to do. "I think the increasing of interest rates and the shrinking of the Fed portfolio is an absolute awful thing to do at this time".
Trumps Finance Minister Steven Mnuchin had tried last weekend to calm the markets.