Dr Fauci, the head of the National Institute of Allergy and Infectious Diseases, appeared on the BBC's Andrew Marr Show on Sunday morning.
The veteran public health expert, the most widely respected in the United States, said that Donald Trump - with whom he has a tricky relationship - was right to say during the debate that a vaccine could be approved by next month.
"I think we will know whether a vaccine is safe and effective by the end of November, the beginning of December," he said.
"The question is, once you have a safe and effective vaccine, or more than one, how can you get it to the people who need it as soon as possible."
Dr Fauci said it will likely take several months to be distributed, but he hoped that the vaccine, coupled with a continuation of public health measures like face masks and social distancing, could help move the US towards “some form of normality” by the middle of next year.
“I think when we get a vaccine, and we start getting people vaccinated, over a period of several months into 2021, we will begin to approach some form of normality, depending upon how many people, what proportion of the people, take the vaccine.
“That should be combined with some degree of public health measures. I don’t think that a vaccine alone right off will get us back to normality."
The 79-year-old, a household name in the US since his pioneering work on the Aids crisis in the the 1980s, said that it was vital world leaders listened to the science, and set a good example.
“I think it's very important. People look at what their leaders say and do, and you can positively or negatively influence behaviour," he said.
“One of the things I’m concerned about in the United States is that part of the anti-science translates maybe into anti-vaccine. Particularly among some of the more vulnerable people, like minorities in our population.
“It would really be a shame if we have a safe and effective vaccine, but a substantial proportion of the people do not want to take the vaccine because they don’t trust authority.
“That would really be unfortunate if that’s the case.”
Asked whether Mr Trump was right to claim the US was “rounding the corner” in its fight against the virus, he replied: “No.”
Saturday saw 83,718 new cases across the US — just 39 cases shy of the all-time record that was reported on Friday.
At least 35 states reported more new Covid-19 cases in the last week than the week prior, according to data from Johns Hopkins University.
Marr then asked whether suggesting that people "inject bleach" was following the science.
"No, of course not - that's obvious," said Dr Fauci.
"I don't want to get into this conversation about pitting me against the president.
"I need to do my job as a public health official.
"With all due respect, I understand what you have to do and what you're asking.
"But to go back and forth pitting me against the president isn't helpful for anything, I don't think. Sorry."
Last week in a phone call to donors, Mr Trump called Dr Fauci an "idiot".
"People are tired of Covid," said Mr Trump. "I have the biggest rallies I've ever had, and we have Covid.
"People are saying whatever. Just leave us alone. They're tired of it. People are tired of hearing Fauci and all these idiots."
He continued: "Fauci is a nice guy. He's been here for 500 years."
The Brooklyn-born doctor laughed off the president's comments.
“Addressing the virus is the only thing I really care about," he said.
"That other stuff, it’s like in The Godfather: nothing personal, strictly business as far as I’m concerned.
“I just want to do my job and take care of the people of this country.”