If NEOM is imagined as Silicon Valley in the Middle East, Qiddiya, the entertainment city, could be its Walt Disney World. A conference exhibition showcasing Qiddiya featured a lion hologram as well as simulated racecar drives and roller coaster rides. Mr. Branson’s Virgin has created renderings of a possible space-tourism site, for which his Galactic space-travel unit would be a potential consultant. Six Flags said last year that it was contemplating construction of a theme park near Riyadh; its current thinking is unclear.

“Right now there’s a great deal of money that leaves the country and goes to different areas,” said Ariel Emanuel, a founder of the talent agency the Endeavor Company, a holding company based in Beverly Hills, Calif., which owns sports, television, fashion and other entertainment businesses. Six months ago, he met with Prince Mohammed in Jidda to discuss the kingdom’s vision for the new city.

“I think we could play a pretty important role kind of in bringing entertainment in many different forms,” said Mr. Emanuel. Hs company is also angling for an investment from the Saudi government fund. “We’re having conversations,” he said.

Mr. Branson’s quick dip happened near the site of the third project, The Red Sea, a location for luxury hotels and ecotourism. Mr. Branson, who owns and lives on Necker Island in the Caribbean, said he was astonished by the animal life during his visit to the Saudi islands, including manta rays and sea turtles laying eggs. He is on the project’s board and is contemplating investing there.

“One of the advantages I suppose you could say of Saudi being sort of not open — not open at all, really, since 1979 — has been that these pristine places have been preserved,” he said. “I want to help give my advice as to how to develop it so that in 50 years’ time, people can go and see it as virgin and beautiful.”