Saudi authorities are demanding at least $6 billion from Saudi Prince al-Waleed bin Talal to free him from detention, people familiar with the matter said, potentially putting the global business empire of one of the world’s richest men at risk.

The 62-year-old prince was one of the dozens of royals, government officials and businesspeople rounded up early last month in a wave of arrests the Saudi government billed as the first volley in Crown Prince Mohammed bin Salman’s campaign against widespread graft.

The...