Mr. Kushner, colleagues say, has kept up a regular work schedule, meeting on Wednesday with Gary D. Cohn, the director of the National Economic Council, to discuss the administration’s impending moves on trade. He is also in touch with Jason D. Greenblatt, Mr. Trump’s Middle East envoy, who is in Israel for meetings with Israeli and Palestinian leaders. And next week, he plans to take part in a high-level economic dialogue with China.

Mr. Kushner and his wife, Ivanka, are not accompanying Mr. Trump to Paris. Instead they plan to attend the annual media conference in Sun Valley, Idaho, sponsored by the investment firm Allen & Company. An official said the couple would pay for their own travel and lodgings.

Mr. Kushner is expected to cooperate in the next several weeks with the Senate and House Intelligence Committees that are looking into Russia’s intervention in the American election and any possible collusion with the Trump campaign. He will have to devote some time to preparing for those appearances with his team of lawyers.

Colleagues of Mr. Kushner said he had remained focused and upbeat despite the drumbeat of negative headlines — a trait they ascribe to his experience dealing with the legal troubles of his father, Charles Kushner, who was convicted of tax evasion and witness tampering.

But even as the White House labors to present a business-as-usual facade, there is evidence that Mr. Trump’s family will be drawn deeper into the investigation. Two officials familiar with the Senate Intelligence Committee’s investigation said the panel was now planning to expand its inquiry to include Donald Trump Jr.

The officials said Mr. Trump’s shifting reasons for the meeting — and his acknowledgment that he was lured by the promise of Russian dirt on Hillary Clinton — had forced the Senate panel to begin examining his role in the campaign, and any contacts he may have had with Russians.

The first step, officials said, would be for Senate investigators to sit down with Mr. Trump. The Senate panel might also request that he turn over emails and financial records from any dealings with Russia, which they have done with other subjects of their investigation.