The coronavirus outbreak has created an awkward, often contentious partnership between President Trump and several governors who have taken potshots at the administration for years.

Mr. Trump and the governors — generally Democrats who have been at odds with the president in the past over immigration, climate change and Obamacare — are making a show of cooperation in the fight against the coronavirus outbreak, but finger-pointing and backbiting continue in California, New York and Washington state.

Washington Gov. Jay Inslee on Sunday attempted to dial down his feud with Mr. Trump over the federal response to the public health crisis while making clear his continued distaste for the president.

“We are very pleased with the federal government helping us right now,” Mr. Inslee said on CBS’s “Face The Nation.” “I don’t care what Donald Trump thinks of me, and I just kind of ignore it.”

Mr. Trump last week called the Democratic governor “a snake” for accepting federal help and then criticizing the effort.

Washington has been hardest hit by the novel coronavirus, which causes a respiratory illness known as COVID-19, with 110 confirmed cases and 16 reported deaths through Sunday.

The outbreak has spread to the East Coast, with cases in Maryland, New York, Pennsylvania and the District of Columbia. The number of infections in the U.S. topped 400 across 33 states on Sunday. The death toll was 21, and all but three victims were in Washington state. As testing increased, the number of infections in the United States climbed above 500.

The impact of the deadly bug is being felt in the District as well: Sen. Ted Cruz, Texas Republican, tweeted Sunday evening that he is undergoing a voluntary self-quarantine because of contact with an infected person at a Washington-area conservative event. Rep. Paul A. Gosar, an Arizona Republican who also met the infected person, made a similar announcement later in the evening. Both men said they had no symptoms.

That followed reports that an infected person attended the recent Conservative Political Action Conference in suburban Washington — an event where President Trump and Vice President Mike Pence spoke. The White House said there was no indication either had met or was in proximity to the infected attendee.

The Trump administration is urging calm but also encouraging Americans to take precautions. It suggests that older people and those with preexisting conditions curtail travel and avoid events with large crowds.

During his failed bid for the Democratic presidential nomination who focused on climate change, Mr. Inslee was a fierce critic of Mr. Trump and even accused him of fostering white supremacism.

After he was contacted last month by Mr. Pence, who is overseeing the federal response, Mr. Inslee said in a post on Twitter that he wanted the administration to stick to the facts about the outbreak.

“I told him our work would be more successful if the Trump administration stuck to the science and told the truth,” the governor tweeted.

On a visit to the Centers for Disease Control and Prevention on Friday, Mr. Trump fired back.

“I told Mike not to be complimentary of that governor because that governor is a snake,” Mr. Trump said. “So Mike may be happy with him, but I’m not, OK?

“If we came up with a cure today, and tomorrow everything is gone, and you went up to this governor — who is, you know, not a good governor, by the way — if you went up to this governor and you said to him, ‘How did Trump do?’ He would say, ‘He did a terrible job.’ It makes no difference,” Mr. Trump said.

Mr. Pence said he and the governor are working together “in a seamless way.”

Mr. Inslee also said state and federal teams are in sync despite what he described as Mr. Trump’s confusing rhetoric about the virus.

“My single focus is and will continue to be the health and well-being of Washingtonians. It’s important for leaders to speak with one voice. I just wish the president and vice president could get on the same page,” he tweeted.

In California, where a cruise ship hit by a COVID-19 outbreak has been kept offshore near Oakland for days, Gov. Gavin Newsom and Mr. Trump have undertaken a political cease-fire.

Mr. Trump said the two had a “good conversation” on the phone Friday.

A positive phone call between the two men is the exception in their relationship. The Republican president and the Democratic governor have clashed since Mr. Newsom took office in January 2019 over homelessness, sanctuary cities and other issues.

The cruise ship saga appeared headed for a conclusion Sunday when it was announced that the Grand Princess would be allowed to dock in Oakland, though passengers would remain aboard at least one more day.

The ship is carrying more than 3,500 people from 54 countries. The Grand Princess was forbidden to dock in San Francisco because of evidence that the vessel was the breeding ground for a cluster of nearly 20 cases that resulted in at least one death after a previous voyage.

Another cruise ship was being held off the coast of Florida while awaiting test results to determine whether two crew members who transferred there from the Grand Princess had contracted the new coronavirus.

The Regal Princess was supposed to dock in Port Everglades on Sunday morning but instead was sailing up and down the coast, the Miami Herald reported. The Coast Guard delivered testing kits Sunday morning, and the Centers for Disease Control and Prevention issued a “no-sail order” for the ship.

New York Gov. Andrew Cuomo, another Democrat who often butts heads with Mr. Trump, took a sharp jab at the president in an appearance on Fox News’ “Sunday Morning Futures.”

“You have the president of the United States stand up and say, ‘Anybody who wants to test can have a test.’ And then you have the vice president stand up and say, ‘We don’t have the capacity to test,’ … that’s what causes the panic and the fear.”

With 105 confirmed cases in the state and nearly 3,000 people in New York City under a precautionary quarantine, the governor slammed the federal government’s response as confused, nonsensical and slow.

He also criticized the state’s share of federal emergency aid, $35 million out of an $8.3 billion measure for the whole nation, as “shocking” and “a drop in the bucket.”

• S.A. Miller contributed to this article, which is based in part on wire service reports.

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