In an interview with BBC Radio 4, Branson said of Bezos: "I think we're both neck and neck as to who will put people into space first. We're talking about months away, not years away — so it's close."
Branson added that he hopes to become an astronaut within 12 months with Virgin Galactic, and he is already preparing for his experience, including gruelling centrifuge training.
Bezos, meanwhile, hopes his company Blue Origin can start offering suborbital space tourism flights by 2019. And in the same week Branson made his comments, he said the space company is "the most important work I am doing."
But while Branson said the imminent prospect of space travel is "exciting," there are reasons to treat his claim with a grain of salt. The billionaire has consistently teased the idea that space tourism is just around the corner, and is yet to deliver on his previous promises.
One of his most recent pledges came in October last year, when he told CNBC that he would be "very disappointed if I haven't been into space within six months or so." Well, six months came and passed in April and Branson is yet to leave Earth's atmosphere.
In fact, British satirical magazine Private Eye has kept track of the times Branson has said space travel is upon us, only for the actual flight not to materialise. It counted 15 separate broken promises by Branson, up to the point when Virgin Galactic's VSS Enterprise crash landed over the Californian desert in October 2014, killing a pilot.
Since then, there have been promising signs for Virgin Galactic. In April, the company completed a supersonic test flight of its SpaceShipTwo passenger rocket ship. And there's no reason to doubt that Bezos will not achieve his aim of sending tourists into space next year after a textbook test flight last month.
But as for Branson's claim that this is all months away, well there's plenty of reasons why you should treat that as a bit of billionaire bravado as the space race really heats up.