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…

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