As the Trump administration has pressed publicly for top-speed development and approval of a coronavirus vaccine, allotting billions of dollars to pharmaceutical companies, political leaders and public health experts have warned of the dangers of rushing the process.

That divide has only grown recently, as two of the country’s high-profile governors, Andrew M. Cuomo of New York and Gavin Newsom of California, revealed their caution about potential vaccines.

Mr. Newsom announced plans to form an independent panel in his state to review any federally approved vaccines before they are administered to residents. “Of course we won’t take anyone’s word for it,” he said in a news briefing Monday.

California’s new case rates have stayed relatively low, but in much of the rest of the country, the numbers are alarming: On Friday, according to a New York Times database, the United States reported at least 70,464 new cases, the highest figure since July 24. Over the past week, there has been an average of 56,655 cases per day, an increase of 30 percent from the average two weeks earlier.

Mr. Newsom’s announcement came after Mr. Cuomo said last month that New York would also review vaccines approved by the federal government — although Mr. Cuomo tied the move to doubts raised when President Trump suggested that he would reject tougher Food and Drug Administration guidelines. “Frankly, I’m not going to trust the federal government’s opinion,” Mr. Cuomo said.

Recent surveys appear to show that the public shares the governors’ skepticism, with the idea of getting a vaccine as soon as it is available losing appeal for many Americans.

In a poll of likely voters conducted by The New York Times and Siena College, 33 percent said they would definitely or probably not take a vaccine after F.D.A. approval.

In a STAT-Harris poll of about 2,000 people, conducted Oct. 7-10 and published Monday, 58 percent of respondents said they would get vaccinated right away, down from 69 percent who said the same in August.

The decline was twice as steep among Black respondents: Just 43 percent said in October that they would get the vaccine, down from 65 percent in August.

Rob Jekielek, the managing director of The Harris Poll, which has been asking the question throughout the pandemic, said two news events appeared to have played a role in the decline: the back-and-forth between the F.D.A. and the White House over vaccine guidelines, and Mr. Trump’s Covid-19 diagnosis and treatment.