Republicans worried on Friday that Mr. Trump would have to remain in the hospital for a significant period of time, imagery they fear would be damaging at a moment when millions of Americans are already voting.

G.O.P. officials were also concerned that a race with very few undecided voters would freeze in place. Multiple party strategists said their polling in the two nights after the presidential debate had revealed substantial slippage, and not just at the top of the ticket.

“This limits Trump’s opportunity to turn this thing around and drive a winning message,” said Terry Sullivan, a Republican consultant. “He’s lost any ability to control the narrative.”

Should the final weeks of the campaign be dominated by the coronavirus, Mr. Trump’s challenge will be intensified by his casual approach to the disease and its deadliness.