However, Sir Richard has not been deterred by Elon Musk's firm, and instead are using SpaceX as a pace setter in the private race to space.
Richard Branson previously said he was "months" away from launching Virgin Galactic's first rocket into space - on Tuesday he said it will happen within "weeks".
"We will be in space with people not too long after that", he added. "And then we will be in space with myself in months and not years", the firm's founder told CNBC. The company is developing commercial spacecraft and aims to provide suborbital spaceflights to space tourists and suborbital launches for space science missions.
But delays and a fatal crash in 2014 prevented Sir Richard's original ambitions.
Branson has invested in space travel since 2004, and had initially expected to fly into to space himself sometime this year.
In an interview with BBC Radio 4's You and Yours programme, Sir Richard said he was receiving astronaut, centrifuge and fitness training. The upper atmosphere is generally regarded to give way to space at an altitude of 62 miles (100km), almost twice the altitude achieved in the July test flight.
When asked about consumer demand for space travel, he told CNBC: "I think the market for people who would love to become astronauts and go to space is very big".
Both companies will offer customers a weightless experience that will last just minutes, passing through the imaginary line marking where space begins - either the Karman line, at 100 kilometres (62 miles), or the 50-mile boundary recognised by the US Air Force.
The mission is planned for 2023, and would be the first lunar journey by humans since 1972.
Virgin Galactic aims to launch the first commercial passenger flight into space, and is in tough competition with US space company Blue Origin. Mr Bezos didn't clarify if he meant crew or fare-paying passengers. Blue Origin said in June that it will start selling passenger tickets for trips into space as early as next year.