During the trip, the prince, who already owns several dozen hotels in Egypt, announced a further $800 million investment in the country’s tourism industry. He drew criticism from some conservative Egyptians for a video that showed him meeting with Egypt’s female minister of investment and international cooperation, Sahar Nasr, aboard his yacht while he was again wearing shorts. It was unusual protocol in a public meeting for a member of the family that rules a hyperconservative Islamic kingdom.

The arrests also come as Crown Prince Mohammed has forged a close relationship with President Trump, who shares his aggressive approach to Saudi’s regional rival, Iran, and his penchant for bold decisions.

By contrast, Prince Alwaleed sparred with Mr. Trump on Twitter during the American presidential election, referring to him as a “disgrace not only to the GOP but to all America.” Mr. Trump fired back, also on Twitter, that he was a “dopey prince” trying to “control our U.S. politicians with daddy’s money.”

But despite his wealth, Prince Alwaleed was not seen as particularly powerful within the Saudi royal family or as a threat to the crown prince’s consolidation of power. His father, Prince Talal, known as the “Red Prince,” spent years in exile after leading a kind of leftist revolt among royals in 1962, and had grumbled in the past about being passed over in the royal succession. Prince Alwaleed himself initially objected to the naming of Mohammed as crown prince, though he quickly stopped complaining in public.

A more likely reason for his inclusion in the arrests, experts said, is that he might have gone bankrupt during the 2008 financial crisis. He had been highly leveraged and somehow got elements of the government to bail him out, through his connections to then-King Abdullah and the finance minister, who is also said to have been arrested. Prince Alwaleed’s son Prince Khaled is married to the minister’s daughter.

“They must have uncovered evidence of irregular activity and wanted to make an example of him,” Ali Shihabi, founder of the independent Arabia Foundation in Washington, said on Sunday from Abu Dhabi in a telephone interview.