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The billionaire's space company Blue Origin looks set to achieve the milestone with its New Shepard space vehicle this year.
A report by CNBC on Thursday (14 January) said the company completed New Shepard's fourteenth test flight of the rocket booster and capsule - one of the last steps before sending its first crew into space.
A spokesperson for the company commented that the news is 'rumor and speculative - not confirmed' when responding to a request for comment by Reuters.
However, the report by CBNC said Thursday's flight was the first of two 'stable configuration' flights, which means the company will avoid making major changes between one test flight and the next.
The second 'stable configuration' test flight is due to be launched within six weeks - or by late February - with the hopes of the first crewed flight being sent up by early April.
In December, Bezos said Blue Origin will take the first woman to the moon's surface.
Bezos said on Instagram: "This (BE-7) is the engine that will take the first woman to the surface of the Moon."
Blue Origin's landing ship is named after his mother Jacklyn.
Writing on Instagram in December, Bezos said: "My siblings and I surprised our mom today. We revealed the name of the @BlueOrigin landing ship. We named it after her - the Jacklyn.
"New Glenn's first stage will come home to the Jacklyn after every flight. It couldn't be more appropriately named - Mom has always given us the best place and best heart to come home to.
"We love you, Mom."
In October last year, Jeff Bezos managed to break a record with his New Shepard rocket ship.
Blue Origin marked the rocket's seventh consecutive flight - in turn breaking the record for the most reused spacecraft.
And with all those previous launches, the rocket has managed to gain itself a bit of a reputation, with many people over the years remarking on how much it looks like a 'penis'.
Responding to the news of the record-breaking launch, one person tweeted: "If I were Jeff Bezos and had 200 million dollars, I would also like to shoot a space penis into the moon and land it in between two craters that look like a butt."
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