President Trump’s announcement on Friday that he had tested positive for the coronavirus set off a wave of tweets and Facebook posts with a common refrain, especially on the left: Why should we believe him?

There was no evidence that Mr. Trump was lying. But overnight, hundreds of tweets were posted casting doubt on whether the president contracted the coronavirus.

The White House has given multiple statements confirming Mr. Trump’s condition. His physician confirmed the positive test result, and Mark Meadows, the president’s chief of staff, said that Mr. Trump had mild symptoms of Covid-19.

The tweets questioning Mr. Trump’s announcement peaked at five per minute on Friday morning according to Dataminr, a social media monitoring service. The doubters included Jelani Cobb, a staff writer at The New Yorker, and Anand Giridharadas, editor at large of Time and an occasional contributor to The New York Times.

Some suggested that the announcement from the president could be an excuse to delay the election, since he is trailing in the polls, and cancel future presidential debates.

Some of the people doubting Mr. Trump said they couldn’t believe him because of how much false and misleading information he has spread about the virus in the past.

Researchers at Cornell University published a study this week showing that Mr. Trump was the single largest driver of false and misleading information about the coronavirus. Mentions of Mr. Trump made up nearly 38 percent of the overall “misinformation conversation,” the researchers said.

Mr. Trump has also stated on at least 34 separate occasions since February that the coronavirus would go away.

“We’re in an environment where conspiracies are thriving, in part because the president encourages them,” said Melissa Ryan, chief executive of Card Strategies, a consulting firm that researches disinformation. “And we have a White House comms operation that gives the press and public disinformation constantly.”

The situation has created “the perfect storm for people to assume that the White House isn’t being truthful,” Ms. Ryan said.

Many of the deniers also latched on to a tweet from Sept. 18 that had originally been shared in conspiracy circles, but was reshared widely on Facebook and Twitter after Mr. Trump’s announcement on Friday. “Trump’s October surprise will be the announcement of ‘his infection,’” it said. “Fake, but quite dramatic.” The post collected nearly 15,000 interactions across Facebook and Twitter, mostly from people who falsely asserted that Mr. Trump catching the virus was a known plan.

And some saw people’s reactions to the announcement as a reflection of the sheer magnitude of misinformation that has emanated from the president.