Speaking to the BBC's "You and Yours" radio show, he said that Tesla CEO Elon Musk is doing "fantastically well" with rocket development and space transport for cargo. But, he added that ultimately the commercial space race was between himself and Bezos.
"I think we're both neck and neck as to who will put people into space first," Branson told the British broadcaster in an interview to be aired Monday.
"Ultimately, we have to do it safely. It's more a race with ourselves to make sure we have the craft that are safe to put people up there."
The serial entrepreneur, who owns the commercial space flight company Virgin Galactic, has invested in space travel since 2004 and was initially expected to go to space himself before April of this year.
Giving an update on his progress to the BBC, he said that there were "exciting times ahead" and his first trip to space would be "months away, not years away."
"I'm going for astronaut training; I'm going for fitness training, centrifuge and other training, so that my body will hopefully cope well when I go to space," he said.
Read more on the BBC's website here.