Gates declined, telling the president that it was "not a good use of my time."
Gates is heavily involved in global health philanthropy through the Bill and Melinda Gates Foundation. Earlier this year, he told USA TODAY he was worried that the U.S. risked losing its geopolitical clout if the Trump administration succeeded in slashing foreign aid, reducing its role in providing aid to poor countries.
The role of science advisor is to help guide the president on issues relating to science and technology, overseeing the Office of Science and Technology Policy.
It is unclear if Trump was serious about the offer, something that Gates concedes.
"I didn't put him to the test, whether that was a serious thing or not. He probably himself didn't know if he was serious. It was a friendly thing. He was being friendly," he told Stat.
The White House did not immediately respond to a USA TODAY request for comment on the offer.
In addition to the potential job offer, Gates also spoke to Trump about a universal flu vaccine. That conversation was probably "the longest conversation about universal flu vaccine that the president's ever had," Gates told Stat.
While it is unclear if the president will act on the idea, Gates recalls Trump being rather interested, so much so that he called FDA Commissioner Scott Gottlieb with Gates in the room to get his perspective.
The Gates Foundation, in concert with the family of Google co-founder Larry Page, announced plans last week for a $12 million fund to help speed up the development of a universal vaccine.