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.