The petition, which was lodged before the Scottish Parliament last week, states that Snowden is owed a "debt of gratitude" for revealing information on the "surveillance" of emails and text messages by British and US security organisations.
It urges the Scottish Government to offer Snowden asylum in the event of a Yes vote, which it says would encourage whistleblowers everywhere to take a stance and signal a "clean break with the intrusiveness of the UK security state".
Fugitive former National Security Agency systems analyst Snowden, who was elected rector of Glasgow University last month, is living in Russia where he has been given temporary asylum. In 2013, US authorities filed criminal charges against him including espionage and theft of government property.
The petition, which has gathered more than 250 signatures, was submitted by Mick Napier, a former university lecturer and campaigner with the Scottish Palestine Solidarity Campaign.
He said: "Edward Snowden deserves this because he has done nothing wrong, he has provided us all with a service. Snowden is the elected rector of Glasgow University by democratic ballot and therefore there is something peculiar about the elected rector of Glasgow University, one of our most prestigious Scottish universities, being a fugitive in Russia.
"Conditional upon a Yes vote, we would like the offer (of asylum in Scotland) to be made right away."
Yvonne Ridley, the former journalist captured by the Taliban, who now lives in the Borders, is among those backing the petition.
A Scottish Government spokeswoman said: "We cannot comment on individual cases.
"Asylum is granted within the terms of relevant international law to those fleeing persecution or serious harm in their own country and in need of international protection, and an independent Scotland would consider each case on its own merits."