“I would think usually you might put a medical professional in charge,” Ms. Klobuchar said, though she acknowledged that the vice president’s involvement might help to elevate the effort. (President Barack Obama named Ron Klain, a Democratic operative, to coordinate the government’s response to the Ebola virus.)

While calling out the president’s proposed cuts to funding for the Centers for Disease Control and Prevention, Ms. Klobuchar also delivered faint praise of Mr. Trump’s choice to invite the agency’s principal deputy director to address the nation alongside him. “He did do it with the C.D.C.,” she said. “I think that’s important, because I believe in science.”

Ms. Warren in recent days has pointed to a plan she released weeks ago focused on combating infectious disease outbreaks. On Wednesday she escalated her proposal, with the specific suggestion of redirecting the border-wall money to address the outbreak.

“We know that with any virus that develops, the most vulnerable will be our children, seniors, people with compromised immune systems,” she said. “This one is tough.”

Mr. Bloomberg, too, planned to step up his message on Thursday, when his campaign will begin airing a new television ad nationally that is focused on the virus. “Health experts warn the U.S. is underprepared,” the ad begins. “Managing a crisis is what Mike Bloomberg does.”

Announcing the commercial Wednesday, Mr. Bloomberg’s campaign tore into the president. “Trump is putting American lives at risk every day, ignoring science, claiming the virus will ‘miraculously’ disappear by April and relying on ‘warm weather’ to end the spread of the virus,” the campaign said in a news release.

On Wednesday, Mr. Trump issued a message on Twitter criticizing the news media for “doing everything possible to make the Caronavirus look as bad as possible, including panicking markets, if possible.” He added, “Likewise their incompetent Do Nothing Democrat comrades are all talk, no action.”