Under the deal, two Saudi astronauts will ride SpaceX's Crew Dragon capsule to the space station for a roughly weeklong stay early next year, the sources said.
Officials with the Saudi Space Commission, Riyadh's space agency founded in 2018, were not immediately available to comment.
The Saudi astronauts will join two previously announced Americans, retired NASA astronaut Peggy Whitson and race car driver and investor John Shoffner, the sources said.
Axiom launched its first private mission to the space station in April, sending a four-man crew to the space station aboard a SpaceX Crew Dragon capsule that included a Canadian investor and an Israeli businessman.
Axiom's astronaut flight business is crucial experience for the company's broader goals of deploying its own private space station by mid-decade.