Questions about Mr. Trump’s handling of crisis, especially in its early days when he suggested it was being used by Democrats to undercut his re-election prospects, are likely to define his presidency. Mr. Navarro’s memo is evidence that some in the upper ranks of the administration had at least considered the possibility of the outbreak turning into something far more serious than Mr. Trump was acknowledging publicly at the time.

Neither Mr. Navarro nor spokespeople for the White House responded to requests for comment.

The memo, which was reviewed by The New York Times, was sent from Mr. Navarro to the National Security Council and then distributed to several officials across the administration, people familiar with the events said. It reached a number of top officials as well as aides to Mick Mulvaney, then the acting chief of staff, they said, but it was unclear whether Mr. Trump saw it.

Mr. Navarro is a well-established China hawk who has long been mistrustful of the country’s government and trade practices. Both Mr. Navarro and Matthew Pottinger, the chief deputy at the National Security Council, were among the few officials urging colleagues in January to take a harder line in relation to the growing threat of the coronavirus.

But their warnings were seen by other officials as primarily reflecting their concerns about China’s behavior — and their concerns look more prescient in hindsight than they actually were, other officials argue.

With the subject line “Impose Travel Ban on China?” Mr. Navarro opened the memo by writing, “If the probability of a pandemic is greater than roughly 1%, a game-theoretic analysis of the coronavirus indicates the clear dominant strategy is an immediate travel ban on China.”

Mr. Navarro concluded at one point: “Regardless of whether the coronavirus proves to be a pandemic-level outbreak, there are certain costs associated with engaging in policies to contain and mitigate the spread of the disease. The most readily available option to contain the spread of the outbreak is to issue a travel ban to and from the source of the outbreak, namely, mainland China.”

He suggested that under an “aggressive” containment scenario, a travel ban may need to last as long as 12 months for proper containment, a duration of time that at that point some White House aides saw as unsustainable.