Both were reported to have submitted unconditional offers of 500m for the outsourcing group, which has sales of around 286m.
PrintWeek revealed in February that Astron chief executive David Mitchell had appointed UBS and Citigroup "to explore our options", which included a sale or flotation.
A City source commented: "Private equity companies are so buoyant at the moment, so it [Astron] would be an obvious one for a private equity company to buy.
"Astron is an interesting business, but the challenge is what more do you do with it to allow Candover or whoever buys it to make a 25% return on the business every year?"
Print giant RR Donnelley was also named as a potential bidder by Financial News Online, although this was deemed unlikely by the City insider.
"I didn't think Astron would go to a printer, as it has really badged itself as support services and document management. I would expect Donnelley to be more interested in the Polestars of this world than Astron," he said.