Dr Moncef Slaoui told US network CNN the plan was to "ship vaccines to the immunisation sites within 24 hours" of a vaccine being approved.
The comments come amid a surge in coronavirus cases across the country.
The US has recorded more than 12m cases and 255,000 deaths, according to data compiled by Johns Hopkins University.
These are the highest tolls registered anywhere in the world.
American pharmaceutical company Pfizer and its partner BioNTech submitted an application on Friday for emergency authorisation in the US of their Covid-19 vaccine.
The vaccine, which requires two doses, has been shown by tests to be 95% effective. Pfizer hopes to produce up to 50 million doses by the end of the year.
A Food and Drug Administration (FDA) vaccine advisory committee is set to meet on 10 December to discuss whether to authorise the vaccine.
Dr Slaoui told CNN it could be rolled out "maybe on day two after approval, on the 11th or the 12th of December".
The vaccine is set to be distributed based on each state's population. Dr Slaoui said individual states will be responsible for deciding who gets the vaccine first, with the recommendation that priority be given to those most at risk, like healthcare workers and the elderly.
He said the country could achieve "herd immunity" in May, with 70% of the population vaccinated.
But he added: "I really hope and look forward to seeing that the level of negative perception of the vaccine decreases and peoples' acceptance increases. That is going to be critical to help us. Most people need to be immunised before we can go back to a normal life."
Dr Anthony Fauci, the US's top infectious disease expert, told BBC partner CBS News that the US could reach herd immunity against Covid-19 "reasonably quickly" next year if enough Americans are vaccinated.
US officials earlier granted emergency authorisation for an experimental antibody treatment, developed by Regeneron, given to President Donald Trump after his Covid-19 diagnosis.
Also on Sunday, G20 leaders said they would "spare no effort" to ensure the fair distribution of coronavirus vaccines worldwide.