It is not clear exactly how Mr. Kushner, who is married to Mr. Trump’s elder daughter, Ivanka, might be deployed. He is exploring the prospects of joining the administration as a formal, but unpaid, adviser. Anti-nepotism statutes prohibit close relatives of the president from playing an active role in the government, but at the same time, the Constitution gives the president broad authority to choose his own advisers. If the Trump administration presses ahead with a Kushner appointment, the matter could wind up in court.

Whatever role Mr. Kushner may play in the administration, he has already had a hand in helping assemble it. Both of Mr. Trump’s most senior advisers, Mr. Priebus, his new chief of staff, and Stephen K. Bannon, his chief strategist, seek Mr. Kushner’s advice routinely, considering his buy-in almost a prerequisite for their proposals to Mr. Trump, two senior Trump officials said. (In deference to Mr. Kushner, the transition team delayed announcing the two men’s appointment until after the Jewish sabbath last weekend.)

“Jared has the trust, confidence and ear of the entire inner circle of the Trump administration, including the most important member of that group, the president-elect,” said Matthew Brooks, the executive director of the Republican Jewish Coalition. He described Mr. Kushner as “one of the most important players right now beyond the president- and vice president-elect.”

Inside the Trump operation, he is known as the staunchest defender of Mr. Trump’s judgment. When Mr. Bannon was recently accused of anti-Semitism and of promoting white supremacist views and conspiracy theories, Mr. Kushner reassured the Trump team. He called Mr. Bannon a man of character and said the widespread criticism was a smear, according to a senior Republican official, speaking about private discussions on the condition of anonymity.

While Mr. Kushner kept himself out of the spotlight throughout the campaign, he has been more visible since the election, walking the White House grounds during Mr. Trump’s recent meeting with President Obama and accompanying the president-elect at a meeting at Trump Tower on Thursday night with the Japanese prime minister, Shinzo Abe.

Mr. Kushner declined to be interviewed for this article. Although he is a newspaper owner and publisher, he has not been quoted in the media since the start of the campaign.