Senior US public health officials have pushed back on Donald Trump’s eagerness to reopen the country quickly, as a senior World Health Organization (WHO) figure warned that Covid-19 “is going to be a virus that stalks the human race for quite a long time to come”.

According to researchers at Johns Hopkins University, by Sunday lunchtime more than 530,000 Covid-19 cases had been confirmed in the US and close to 21,000 deaths, making the US the country with the most cases and the most fatalities.

In New York, the worst-hit state by far, Governor Andrew Cuomo reported 758 more deaths on Saturday, raising the toll to 9,385, more than three times the state’s death toll from the 9/11 attacks.

Cuomo reported signs of progress including a flattening number of hospitalisations but said the daily death rate, also flat but high, was “terrible news”.

“That’s the one number I look forward to seeing drop,” he said.

On Saturday night, Trump said a decision to open up the economy was one he alone would make, and would be “the biggest” of his presidency. He has targeted 1 May as the date when the country may begin a return to normalcy, and in a tweet on Sunday morning cited a drop in hospitalisations as “a very good sign”.

But Dr Anthony Fauci, the nation’s leading infectious diseases expert, stressed any such move should come only when it was safe and as a gradual process. He also warned of the danger of a resurgence of Covid-19 cases later in the year if things reopened too soon.

“If you just say, ‘OK, it’s whatever, 1 May, click, turn the switch on,’ obviously, if you do it in an all-or-none way, there’s an extraordinary risk of there being a rebound,” he told CNN’s State of the Union. “That could be a real problem. And everybody knows that.

“When one starts to relax some of those restrictions, we know that there will be people who will be getting infected. That is just reality. The critical issue is to be able to, in real time, identify, isolate and contact trace.

“Obviously, New York, [which] is going through a terrible ordeal, is going to be very different from Arkansas, and very different maybe from some places on the west coast, like Washington state, which have been successfully able to prevent that big spike. I think it’s going to have to be something that is not one size fits all.”

Fauci was backed by Dr Stephen Hahn, commissioner of the Food and Drug administration (FDA).

“The primary issue here is the safety and the welfare of the American people. That has to come first,” Hahn told NBC’s Meet the Press, noting he had not felt political pressure on any decision. “Testing is one component of the response to the outbreak, in addition to the mitigation effort … so all hands on deck to try to get more diagnostic tests in.”

WHO special envoy Dr David Nabarro made a grim prediction.

“We think it is going to be a virus that stalks the human race for quite a long time to come until we can all have a vaccine that will protect us and that there will be small outbreaks that will emerge sporadically and they will break through our defences,” he said, also on NBC.

Trump has been highly critical of the WHO, in comments some have seen as scapegoating to deflect from the failures of his own administration.

“The WHO works on behalf of all the governments in the world and it operates within mandates that are given to it,” Nabarro said. “We know there will be many things found to have perhaps not been done as well as they could have been, and we’re anticipating there’ll be lots of examinations afterwards. Right now, we have to move forward.”

Trump’s Sunday tweet appeared to indicate he was still keen to reopen the country sooner than later, having told Fox News on Saturday night he would seek advice from “very smart people”. Last week, Trump announced he would establish a bipartisan “Opening Our Country Council”.

“A lot of very smart people, a lot of professionals, doctors and business leaders are a lot of things that go into a decision like that,” he said. “And it’s going to be based on a lot of facts and a lot of instinct also. Whether we like it or not, there is a certain instinct to it.”

Among governors of hard-hit states, instincts tend towards caution. Phil Murphy of New Jersey told CNN 2,183 “blessed souls” had died in his state and repeated to CBS’s Face the Nation: “If we start to get back on our feet too soon … we could be throwing gasoline on the fire.”

Trump’s Fox phone-in came ahead of another tweet in which he attacked the New York Times, which on Saturday reported that the White House failed to act on several explicit security agency warnings in January about the potential for coronavirus to spread to the US.

Trump conflated reports on the outbreak and mis-represented Times reporting which said researchers think most Covid-19 cases in New York came from Europe – not that the virus had originated there.

“So now the Fake News is tracing the CoronaVirus origins back to Europe, NOT China,” Trump wrote. “This is a first! I wonder what the Failing New York Times got for this one? Are there any NAMED sources?”

On Wednesday, for example, The Times quoted Harm van Bakel, a geneticist at Icahn School of Medicine at Mount Sinai, who co-wrote a study awaiting peer review, as saying: “The majority [of New York cases] is clearly European [in origin].”

Trump also claimed the Times was “recently thrown out of China like dogs, and obviously want back in. Sad!”

Times journalists were among Americans expelled by the Chinese government in response to the Trump administration limiting the numbers of Chinese citizens who can work for Chinese news organisations in the US.