Meadows said the Trump administration's goal is to have 100 million doses in an initial tranche, which would primarily go to individuals most at risk of contracting the virus.
"We believe that we can get the vast majority of those at risk with a distribution model that would be available at the end of October," he told reporters outside the White House.
The second tranche of 200 million-plus could be ready to distribute by January 2021, he added.
Both President Donald Trump and Vice President Mike Pence said on Sept. 16 the administration was on track to have 100 million doses ready by the end of the year.
"We’ll be able to distribute at least 100 million vaccine doses by the end of 2020 and a large number much sooner than that," Trump told reporters in Washington.
The vaccination campaign is to begin gradually in January or possibly later this year, according to the documents.
Scientists are racing to test vaccine candidates, with a number in phase three, the last phase required before approval and distribution.
Through Operation Warp Speed, a Trump administration program, some candidates are already being manufactured to speed the timeline in case they're approved by the Food and Drug Administration (FDA).
"So the very moment that one of those companies that are already in phase three clinical trials says that it’s safe, they submit it to the FDA, as soon as the FDA says it’s safe and effective, we will have tens of millions of doses available for the American people," Pence said.
Some vaccines being produced will be discarded, Meadows said, because they won't have been proven safe and effective.
Concerns are rampant that this would be the fastest vaccine produced in history, with many Democrats accusing Trump of trying to rush development to get one before the Nov. 3 presidential election, and others dismissing hopes that it will be available soon.
"Even if we get a vaccine, it will not be available for most of the population until well into 2021," Democratic presidential nominee Joe Biden said during a speech this week.
Vaccine developments, he said, "don’t adhere to election cycles, and their timing, their approval, and their distribution should never, ever be distorted by political considerations."
Trump has in turn accused Biden and others of promoting "anti-vaccine theories."
Asked about the criticism and polls, Meadows told reporters: "We can look at all kinds of polls, and I don't think you, as a member of the press, want to look at the polls in terms of how they view you."
"Let's just be serious about it. You're going to have contradictory information all the time. I can tell you what the president has been consistent with—and he's consistent with me last night, late on the phone—is making sure that every single minute of every single day is there to develop a cure, in this case both therapeutics and vaccines, to make sure that we tackle this virus that none of us had ever heard of until January of this year."
Centers for Disease Control and Prevention (CDC) Director Robert Redfield, meanwhile, told members of Congress earlier Sept. 16 that normal life wouldn't return until next year.
Addressing when "we can begin to take advantage of vaccine to get back to our regular life, I think we're probably looking at late second quarter, third quarter 2021," he said.
Redfield later said he was talking about when everybody would be able to get vaccinated.
"While Dr. Redfield may have a timeline in mind, to my knowledge he hasn't had intimate discussions with those processes," Meadows told reporters, noting that the CDC isn't directly involved with vaccine development. "That timeline is not consistent with what I have had personal interaction with."