In an interview with CNBC, Branson said his company was "more than tantalisingly close" to its first landmark flight beyond Earth's atmosphere, stating: "we should be in space within weeks, not months". "Then we will be in space with myself in months and not years".
A ticket into space on board a Virgin Galactic flight will be a costly £192,000 ($250,000) - but "ultimately" he would expect it to fall to £31,000 ($40,000) within a decade.
The 67-year-old multi-millionaire was initially expected to take part in a space flight before April this year.
"We will be in space with people not too long after that so we have got a very, very exciting couple of months ahead".
Branson has been working out and going through astronaut training and centrifuge training in recent months to prepare for making the journey himself.
Virgin Galactic's SpaceShipTwo on its first test flight in October 2010 to prove the glide-and-land design.
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The British billionaire said Musk's late-night tweets - which have landed him in hot water a number of times this year - are a key "flaw" he must overcome.
Talking to the BBC at that time, Branson said: "I think we're both neck and neck as to who will put people into space first".
He founded the commercial spaceflight company in 2004.
The reusable New Shepard rocket and spacecraft is meant to carry up to six space tourists, researchers and/or experiments on brief suborbital flights, the company has said.
Virgin previously said that it had sold around 650 tickets for its fully commercial suborbital flights, which are now booked until at least 2021.
"If I have a room full of 10 people, eight out of 10 would love to go to space if they could afford it", Sir Richard told CNBC.
But Virgin boss Richard Branson has come out with some bullish words on Galactic's progress. "And it is up to us to produce as many spaceships as we can to cater with that demand".