The Amazon CEO announced this week that he and his brother would be heading to space on July 20 on Blue Origin's New Shepherd spacecraft.
However, according to a report by Doug Messier on the Parabolic Arc website, "Virgin Galactic is working on a plan to send Branson on a suborbital flight aboard the VSS Unity SpaceShipTwo rocket plane over the July 4 holiday weekend."
If that's true, Branson would beat Bezos to space by 16 days.
Messier said his source wishes to remain anonymous but the US space journalist has been connected to credible sources in the past.
Branson's Virgin Galactic space company has neither confirmed nor denied the rumour.
When asked by Ars Technica whether the report was true, Virgin Galactic said: "We are in the process of analyzing the data from our successful May 22nd flight.