Secret service and regular police have been searching for Ivan Kaspersky, 20, for at least two days, a police source told Interfax news agency.
His father's firm, Kaspersky Lab, told a newspaper it could not confirm news he had been abducted.
Yevgeny Kaspersky made his fortune developing anti-virus software.
News of his son's abduction was reported by Russian news website Life News, quoting its own sources.
The kidnappers are demanding 3m euros (Â£2.6m; $4.3m) for Ivan's safe return, the website says.
According to the unconfirmed report, he was snatched while on his way to work in Moscow on Tuesday.
When contacted by Russian newspaper Gazeta, Kaspersky Lab said it could neither confirm nor deny the report.
There has been no official comment on the story.
Kaspersky Lab is regarded outside Russia as one of the country's few business success stories not related to the energy sector.
The US business magazine Fast Company recently ranked Kaspersky Lab among the Top 50 Most Innovative Companies worldwide.
The son of Russian software entrepreneur Yevgeny Kaspersky has gone missing in Moscow and may have been kidnapped, Russian media report.