The 15-year-old rapper was eliminated from the Fox singing competition last week after polling the fewest number of public votes for his performance of 'Black or White' on the Michael Jackson-themed performance show.
Speaking on Access Hollywood Live following his exit, the teenager namechecked Jay-Z and Kanye West as "the top two people" he admired in the industry.
When Billy Bush quizzed if he wanted to be a businessman like Jay-Z, Astro quipped: "I want to be a better businessman than Jay-Z. It will take time but it can happen."
As for whether he has heard from either Jay-Z or Kanye, he said: "No, I haven't heard from them. It's weird because they always come around me for some reason but I never get to meet them."
Astro then recounted a time when his mother walked past Jay-Z but didn't say anything for fear of being ignored, joking: "I missed out on a huge opportunity. He could have known me from then!"
Watch Astro talking to Access Hollywood Live below:
Astro was mentored by record executive LA Reid on The X Factor, which also said goodbye to 14-year-old singer Drew Ryniewicz in last week's double elimination.
Reid has since said that Astro has already proven himself a success on the show and predicted that the public will be hearing his original songs again in the future.
However, Astro recently denied reports that he had already been signed by Jay-Z's label Roc Nation, adding that whatever happens in his future career his main goal is to keep having fun with his music.
"I just want to be able to have fun doing it," he told MTV. "I don't want music to be a job for me; I want it to be something I'm doing for fun. As long it's not a job then you're straight."
The acts still in the running for the $5 million recording contract prize are: Melanie Amaro, Rachel Crow, Josh Krajcik and Marcus Canty.
X Factor continues Wednesdays and Thursdays at 8/7c on Fox.
> Timbaland sends support to X Factor Astro after elimination
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Watch Astro perform 'Black or White' below: