In the first months of the administration, Mr. Kushner typically would spend five or six hours a day with the president in what his advocates described as playing defense, making sure others were not gaming the system by persuading Mr. Trump to make decisions without consulting others who had interest in the issues. Now under a less freewheeling system, Mr. Kushner and other aides are expected to stay in their own lanes.

Critics take a less generous view. Rick Tyler, a Republican strategist, said the “magical powers” ascribed to Mr. Kushner early on seemed to have faded. “As long as Jared was seen and not heard, he was able to play the role of wonder boy,” he said. “But now he is no longer seen, and we are only left to wonder about the boy whose father-in-law placed the hope of unraveling the world’s most intractable public policy puzzles from peace in the Middle East to reinventing government” in him.

Speculation about Mr. Kushner’s role comes as the long shadow of the special counsel’s investigation darkens the White House. Investigators have asked witnesses about Mr. Kushner’s foreign policy role during the campaign and the presidential transition, including his involvement in a debate over a United Nations Security Council resolution condemning Israel’s construction of settlements in the occupied West Bank, as The Wall Street Journal has reported.

One person familiar with the questioning, who asked not to be identified discussing the investigation, said the special counsel, Robert S. Mueller III, appeared to be exploring Mr. Kushner’s role as part of his examination of Michael T. Flynn, who went on to become Mr. Trump’s national security adviser before being forced out after 24 days for not being forthcoming about his contacts with Russia’s ambassador.

Mr. Kushner has yet to be interviewed by Mr. Mueller’s team. Worried that his conversations might have been picked up on a government-authorized wiretap or perhaps by Russia or China, Mr. Kushner has become increasingly cautious about how he communicates, even with friends.

The scrutiny by Mr. Mueller upended Mr. Kushner’s initial hopes, according to some colleagues. Mr. Kushner expressed relief over Mr. Mueller’s appointment in May, assuming that the prosecutor’s inquiry would effectively freeze congressional investigations and therefore free up the White House to pursue its legislative agenda.

To some, that suggested he did not get it, that he did not fully grasp how the special counsel would scrutinize every single thing he had done in business, during the transition and during the campaign. And the emergence of Mr. Mueller has not halted the House or Senate inquiries; Mr. Kushner has since been interviewed by congressional investigators.