Virgin Galactic, which charges US$250,000 a ticket, has some 700 people signed up to fly, and Branson has said he would be among the first to go.
To prepare for his flight, which he has said could come this year, the 67-year-old said he's been cycling, playing tennis in the morning and evening, and spending time in a centrifuge to get his body used to the additional gravitational forces passengers would experience on SpaceShipTwo.
Blue Origin, the space company owned by Jeff Bezos, is also aiming to fly its first test flights with people by the end of this year.
Branson said he expected that the companies would both "have a person in space roundabout the same time." But he said they "are not in a race to get to space. . . . All that matters in the end is that everybody is safe and well."
He said that some of his customers have also expressed interest in flying with Blue Origin, which has not yet set a price. "The more spaceships that get built, the better the price will be, and the bigger the market, and the more resources that Jeff and ourselves will have to invest in exciting things going forward in space," he said.
SpaceShipTwo is "air launched," meaning it is tethered to the belly of a mother ship, which flies to some 45,000 feet (13,715m). Then the spacecraft is released; it fires its engine and powers off through the atmosphere.
Branson has been attempting to get to space for years, ever since he acquired the rights to the technology of the spacecraft from Paul Allen, the co-founder of Microsoft who in 2004 backed a venture that flew a vehicle past the 100km edge of space three times.
Since then, Branson has been pursuing his own quest to build an even bigger vehicle that would be capable of carrying as many as six passengers and two pilots to the edge of space, where they would enjoy floating weightlessly around the cabin and views of the Earth from above.
But the programme has had multiple setbacks. The schedule has been delayed for years. And in 2014, a previous version of SpaceShipTwo came apart midflight, killing the co-pilot, Michael Alsbury.