The Mayo Clinic and Cambridge-based biotech company nference, found in the study that both mRNA COVID-19 vaccines' effectiveness dropped in the month of July. The researchers, while saying the vaccines provided good protection against the virus, found that Moderna's vaccine effectiveness was 76 percent, as compared with Pfizer's 42 percent.

"That study … is a pre-print study, it hasn't been fully peer-reviewed," Fauci said on CBS's "Face the Nation" of the recent Mayo-nference study's findings, which was published on the website medRxiv.org. The study gathered data from about 25,000 Minnesota residents from January and July.

"I don't doubt what they're seeing, but there are a lot of confounding variables in there, about when one was started, the relative amount of people in that cohort who were Delta versus Alpha," he said, referring to two COVID-19 variants. He didn't elaborate.

Both the Moderna and Pfizer vaccine should be used as booster shots, he said.

"Right now, if we get boosters … it's clear we want to make sure we get people, if possible, to get the boost from the original vaccine," Fauci said.

Fauci has been the head of the National Institute of Allergy and Infectious Diseases since 1984 and has become one of the federal government's public faces in its messaging about COVID-19. In the interview, Fauci also said he hopes the Food and Drug Administration (FDA) fully authorizes the vaccines.