WASHINGTON — It started as a series of conversations this past week between officials working with Jared Kushner, the president’s son-in-law, and the chief executive of Verily, a life sciences subsidiary of Google’s parent company, about how it might help the Trump administration in the fight against the coronavirus.

Verily was developing a website that could let people evaluate their symptoms and direct them to nearby “drive through” locations for testing. Desperate to tap the private sector to satisfy the public’s demands for a more robust response to the rapidly spreading virus, Mr. Kushner was quickly sold on the idea.

But on Friday, President Trump inflated the concept far beyond reality. At a news conference in the Rose Garden, he said that the company was helping to develop a website that would sharply expand testing for the virus, falsely claiming that “Google has 1,700 engineers working on this right now” and adding that “they’ve made tremendous progress.”

In truth, the project at Verily — which has a total of about 1,000 employees — is in its infancy. A pilot program is planned for the San Francisco area, but a website has yet to be unveiled. Testing locations have not been identified, and the coronavirus tests themselves are not yet widely available.

The president’s effort to sell the website as a significant response to an urgent public health crisis came amid a national outcry over the administration’s repeated failures to deliver on promises to quickly expand access to testing for the virus. The disconnect between Mr. Trump’s exuberant comments and the project’s more modest expectations was the latest example of the president exaggerating, overselling or making wholly inaccurate statements about his administration’s response to the virus, even as facts on the ground contradicted his rosy assessments of progress.

In the hours after the president’s remarks, Google officials distanced themselves from the project, directing question to Verily, where a spokeswoman said the company was merely “in the early stages of development.” The 1,700 engineers Mr. Trump mentioned were actually just Google employees who said a day earlier that they would be happy to volunteer their time on the project if needed.

Now, with Google executives eager to show they are working with the president, the company is racing to meet the promise even as they acknowledge that the debut of the website will be far more limited than Mr. Trump has suggested. A person familiar with the project said Verily was planning to start on Monday with a website that could direct people in the San Francisco Bay Area to as many as three virus testing locations.

Vice President Mike Pence told reporters on Saturday that “the objective here is to have a website up very quickly,” and promised that more details about the partnership with Google would be revealed on Sunday. He said the website would initially be used “in the areas that have been deeply impacted — Washington State, California, New York.”

Mr. Trump has repeatedly made statements about efforts to combat the virus that are wrong. Even as testing was severely limited, he said that “anyone who wants a test can get one.” He suggested the country was “very close to a vaccine,” despite evidence from health experts that one is probably at least a year away. And he stated he “closed” travel from Europe even though Americans and legal residents are exempted.

This account of the president’s embrace of a Google-backed website as a tool for fighting the virus is based on interviews with administration officials and Google employees familiar with the project, and the communications between the company and the White House.

In the past 10 days, Mr. Kushner, who serves as a senior adviser to the president, has taken an increasingly central role in Mr. Trump’s response to the virus. He helped draft the president’s prime-time address this week from the Oval Office, which drew widespread criticism for its repeated errors and misleading statements. He has also reached out to Dr. Kurt Kloss, a physician based in New York, for advice. Dr. Kloss is the father of Karlie Kloss, who is married to Joshua Kushner, Mr. Kushner’s brother.

Administration officials said people close to Mr. Kushner had been in touch with Andy Conrad, the chief executive at Verily, earlier in the week about whether the subsidiary of Alphabet, Google’s parent company, could somehow help with testing. Mr. Conrad, who is known within Alphabet as an executive with an independent streak, described the yet unnamed website to them.

Mr. Kushner, who has led the government’s outreach to the technology community, liked the idea, according to one administration official. He directed his team to work with Mr. Conrad and others at Verily on how their original project might be adapted to help fight the virus.

At that point, Sundar Pichai, the chief executive of Alphabet, was not involved in the discussions, according to people familiar with the situation. Mr. Pichai contacted Mr. Conrad on Wednesday, after an employee asked a question during one of the company’s all-staff meetings about whether Verily could help with testing.

In a memo to his staff the next day, Mr. Pichai wrote that “a planning effort” was underway at Verily to use its expertise in clinical research to aid in the testing.

“As more test kits become available, the planners are looking to develop a pathway for public health and health care agencies,” Mr. Pichai wrote, adding that the effort was targeted at making sure that “individuals who are at higher risk can be directed to testing sites based on the latest guidance from public health authorities.”

In his Thursday memo, Mr. Pichai noted that Verily is “part of our Alphabet family” and could assist in the coming “days and weeks” to respond quickly. He attached a link to a form where Google employees could volunteer to help. Over the next day, more than 1,700 signed up, a number that was communicated to Mr. Kushner’s team.

On Friday, reeling from the collapse of the stock markets and warnings by health experts of the virus’s rapid spread, Mr. Trump scheduled an afternoon news conference.

In the hours before the president’s appearance, Mr. Kushner’s team prepared a poster showing how the website might work; it did not mention Google specifically, but a flow chart showed that a “screening website” could provide “new options for consumers” around the country as they seek to find out whether they are infected.

During the news conference, Mr. Trump was enthusiastic.

“I want to thank Google — Google is helping to develop a website,” he said. “It’s going to be very quickly done, unlike websites of the past, to determine whether a test is warranted and to facilitate testing at a nearby convenient location.”

Moments later, at Mr. Trump’s invitation, Dr. Deborah L. Birx, who is coordinating the response to the virus, held up Mr. Kushner’s poster, saying that it represented a “new approach to testing, which will start in the screening website up here, facilitated by Google.”

The initial reaction 3,000 miles away in Silicon Valley was swift.

Mr. Trump’s comments caught Google completely off guard and executives rushed to issue a statement on Friday afternoon to temper expectations, making it clear that the website is an effort led by Verily, not Google itself. Carolyn Wang, Verily’s spokeswoman, said the plan was to start a pilot website for testing locations in the Bay Area but offered no timetable for a broader rollout.

On Friday evening, Mr. Kushner and Mr. Pichai spoke about the website for the first time, according to an administration official.

Since then, Google executives have scrambled to coordinate with the White House in an effort to do as much as possible to make the president’s vision for the website a reality. On Saturday, Google placed a link on its home page — a rare use of its digital canvass — to encourage people to take steps to prevent spreading the virus.

In addition, Google said it was working on a separate “national informational website” to provide information on virus symptoms and testing sites.

In a statement, company executives wrote that they are “aligned with the U.S. government to work to contain the spread of the Covid-19 virus and to protect the health of our communities.”

Administration officials on Saturday sought to play down expectations, saying that the website could provide significant help in the fight against the virus if it operated only in the nine biggest hot spots around the country. One official noted that Mr. Trump never said that the website would operate in every community nationwide.

The administration is working to make quick testing available at drive-in locations like the parking lots in front of Walgreens pharmacies or Target stores. The Verily website could help drive patients to those locations, officials said.

Mr. Kushner declined to comment.