While much of the U.S. economy has ground to a halt because of the coronavirus outbreak, several dozen workers in orange vests and hard hats were hauling heavy equipment on Sunday at a General Motors plant in Kokomo, Ind.

The crew was part of a crash effort to make tens of thousands of ventilators, the lifesaving machines that keep critically ill patients breathing. The machines are in desperate demand as hospitals face the prospect of dire shortages. New York State alone may need 30,000 or more.

President Trump on Friday accused G.M. and its chief executive, Mary T. Barra, of dragging their feet on the project and directed his administration to force the company to make ventilators under a 1950s law. But accounts from five people with knowledge of the automaker’s plans depict an attempt by G.M. and its partner, Ventec Life Systems, a small maker of ventilators, to accelerate production of the devices.

With deaths surging as cases snowball, the two companies have moved urgently to find parts, place orders and deploy workers, the people said. Tasks that normally would take weeks or months have been completed in days. The companies expect production to begin in three weeks and the first ventilators to ship before the end of April.

On March 19, G.M. began collaborating with Ventec, which normally makes about 200 machines a month, to figure out how to make about 10 times as many in that time. Working through the weekend of March 21 and 22, they hurried to find new suppliers that could provide parts in high volumes, said the five people, who asked not to be named because they fear it would further antagonize Mr. Trump.

Over the weekend, G.M. called in workers to clear out the Kokomo plant, which has been idled because of the outbreak, of machinery previously used to make electrical components for cars. Over the next few days, the automaker and Ventec plan to begin setting up an assembly line. G.M. is already taking applications for the hundreds of jobs. “We continue to work around the clock on our efforts with Ventec,” G.M. said in a statement on Sunday night. “We are working as fast as we can to begin production in Kokomo.”

“I’m pretty amazed at what they’ve done,” said Kristin Dziczek, vice president for industry and economics at the Center for Automotive Research in Ann Arbor, Mich. “But automotive production involves a massive supply chain, and G.M. has risen to the occasion on other big manufacturing challenges.”

Mr. Trump does not see it that way. On Friday, he said on Twitter that Ms. Barra and G.M. had promised to provide 40,000 ventilators “very quickly” but was now telling the administration that it could produce only 6,000 by late April and wanted “top dollar” for the machines. “Always a mess with Mary B,” he said.

G.M. wasn’t negotiating price and other details with the government. Ventec has led the talks with the Federal Emergency Management Agency and the Department of Health and Human Services about how many ventilators the government would like to buy, and at what price. The automaker has said it will not make a profit on the ventilators it assembles and is only seeking to cover its costs.

G.M.’s involvement in ventilators began with a phone call asking for help.

This month, Ms. Barra was contacted by a representative of Stop the Spread, a nonprofit campaign started by Rachel Romer Carlson, the chief executive of Guild Education, and Kenneth I. Chenault, the chairman and managing director at the venture capital firm General Catalyst and a former chief executive and chairman of American Express, four people familiar with the discussion said.

The two wrote an opinion essay on March 18 in The New York Times calling on corporate executives to join the fight against the pandemic. Some 1,500 corporate executives have signed a letter pledging to help in response to their plea.

The four people familiar with the call said that Ms. Barra offered to help and the representative from Stop the Spread suggested the company ought to use its manufacturing and purchasing might to help Ventec scale up ventilator production.

Ventec isn’t a giant in the ventilator industry, but it is known for its VOCSN model, which received F.D.A. approval in 2017. The VOCSN, which is the size of a large toaster oven, combines a number of functions that had previously been performed by several machines to pump air into the lungs, suction out secretions and produce oxygen when a central oxygen line is not available. The device is used in critical-care hospital units but also can be used for home care.

Around the time of the call, G.M., Ford and Fiat Chrysler were grappling with whether to keep their plants running. The United Auto Workers union was putting pressure on the companies to do more to protect workers. The day after Ms. Barra spoke to Stop the Spread, G.M. and the other two large U.S. automakers said they would shut down plants to until at least March 30.

The following day, Phil Kienle, G.M.’s head of manufacturing for North America, and a few other executives flew to Ventec’s headquarters in Bothell, Wash. Early on Friday, March 20, the G.M. team sat down with Ventec executives to learn how the ventilators are made, and what parts are required. Ventec had already started a push to ramp up production to 1,000 a month. The group concluded that with G.M.’s resources, 20,000 a month would be possible, four people familiar with the talks said.

The next day, G.M. emailed its suppliers specifications of Ventec parts, asking if any could produce them in high volumes. Mr. Kienle’s team quickly zeroed in on Kokomo as a location to assemble the machines, a person familiar with the matter said. The plant, unlike much grittier car assembly factories, has the type of clean room needed for making medical devices.

On the evening of Sunday, March 22, G.M.’s purchasing chief, Shilpan Amin, emailed Ms. Barr and other top executives to let them know that the company and Ventec had secured commitments from suppliers to produce 95 percent of the needed parts, according to three people familiar with the email.

By last Tuesday, G.M. and Ventec had the details of their collaboration hammered out, which they discussed publicly early in the week. G.M. would operate as a contract manufacturer for Ventec, which would sell and distribute the machines. Ventec also plans to increase production at its plant in Washington State and has already begun hiring 1,000 additional workers.

As talks progressed, coronavirus cases were soaring in New York City, and climbing as well in Boston, Detroit, New Orleans and elsewhere. In a news conference, Gov. Andrew M. Cuomo of New York complained that the federal government had provided only 400 ventilators to the state. “You pick the 26,000 people who are going to die because you only sent 400 ventilators,” Mr. Cuomo said last Tuesday.

Two days later, Mr. Trump disputed the governor’s numbers while calling in to Sean Hannity’s show on Fox News. “I don’t believe you need 40,000 or 30,000 ventilators,” Mr. Trump said. “You go into major hospitals sometimes, and they’ll have two ventilators.”

The federal government hasn’t indicated how many machines the ventilator makers, including G.M. and Ventec, ought to produce, two people familiar with the talks said. Ventec never received a confirmation from the government about which machine it was interested in acquiring, how many it wanted and how much it was willing to spend.

At the same time, administration officials told The Times that they were struggling to understand how many ventilators the two companies could produce. On Wednesday afternoon, FEMA told the White House that it needed more time to assess offers for ventilators.

When Mr. Trump lashed out at G.M. on Friday, executives at both companies were stunned. G.M. executives were furious Mr. Trump would attack the company after it had made so much progress in a week and the administration had earlier been supportive of their effort. (The president on Friday also took aim at Ford Motor, writing, “FORD, GET GOING ON VENTILATORS, FAST!!!!!” Ford is helping General Electric’s health care division increase production of its ventilators.)

“What we’ve accomplished in five days is incredible,” Larryson Foltran, who works in a technology support group at G.M., wrote on Facebook, noting he had been working 14 to 18 hours a day. He said that the president’s posts had bothered him “on a deeper level.”

Ultimately, G.M. and Ventec executives decided that they would offer no direct response to the president because responding would only invite more criticism from the White House, two people familiar with those discussions said.

Even if the federal government ultimately declines to buy the machines Ventec and G.M. make, the companies are moving ahead because they know there will be other customers around the country, and across the world, four people familiar with their plans said.

But Mr. Trump’s Twitter posts appear to have unnerved other corporate leaders. A person familiar with the Stop the Spread campaign said that several corporate executives who had been willing to contribute to the effort earlier had backed away for fear of ending up becoming targets for Mr. Trump as Ms. Barra had.

At a White House news conference on Sunday, the president struck a different tone. “General Motors is doing a fantastic job,” Mr. Trump said. “I don’t think we have to worry about General Motors now.”