“They were happy to see Obama go,” Bruce Riedel, a senior fellow at the Brookings Institution, said of the Saudis. Mr. Riedel said the kingdom had lost confidence in Mr. Obama after the Arab Spring swept across several countries in the Middle East and North Africa in 2011 and because his administration often pressured Middle Eastern leaders, such as President Abdel Fattah el-Sisi of Egypt, on human rights concerns.

“Trump has made it clear he is not worried about supporting human rights or freedom; he’s made clear that Sisi is going to be his best friend in Egypt; that all those difficult questions about gender equality and the like are going to be off the table for the next four years, and that Iran is very much on the table,” Mr. Riedel said. “As the Saudis look at Trump, they see they don’t need to worry about any of that.”

Still, like many other leaders around the world, the Saudis view Mr. Trump with some degree of wariness, uncertain about the basic competency of his administration and eager to size up a president who has no experience in handling geopolitical affairs. Mr. Trump also sent some mixed signals to the Islamic world in the opening days of his presidency, including in signing a travel ban targeting predominantly Muslim countries, which excluded Saudi Arabia but was widely regarded as the fulfillment of a campaign promise to enact a “Muslim ban.”

Mr. Riedel said the Trump administration — and particularly Jim Mattis, the secretary of defense — “recognizes that we need to clarify that signal with the Saudis, and the best way to do it is with the king’s favorite son.”

Prince Mohammed, 31, is second in line to the throne. He oversees Saudi Aramco, the state-owned oil company, and serves as defense minister, putting him in charge of the Saudi-led intervention in Yemen. Saudi officials see the Houthis in Yemen as a national threat and would like greater American assistance in the fight against them. Saudi Arabia is a major buyer of American weapons.

Prince Mohammed is also the guiding force behind a plan, known as Vision 2030, to transform the kingdom and reduce its dependence on oil. While Prince Mohammed is in Washington, his father, King Salman, is touring Asia in a trip aimed at attracting foreign investment to the kingdom.

Mr. Trump had been scheduled to spend much of his day on Tuesday with Chancellor Angela Merkel of Germany. A snowstorm that blanketed much of the northeastern United States prompted Ms. Merkel to delay her visit until Friday, leaving Mr. Trump’s lunch hour available for Prince Mohammed.

Joining the president for the meetings were Vice President Mike Pence; H. R. McMaster, the national security adviser; Stephen K. Bannon, the president’s chief strategist; Reince Priebus, the White House chief of staff; and Mr. Kushner. Mr. Mattis is expected to meet with Prince Mohammed later this week.