The Kremlin says the move is for data protection but critics fear it is aimed at muzzling social networks like Twitter and Facebook.
The Russian government is thought to be seeking greater access to user data.
Social networks were widely used by protesters opposing President Vladimir Putin's return to the Kremlin in 2012.
Analysts say there are fears that Russia may be seeking to create a closed and censored version of the internet within its borders.
The new bill must still be approved by the upper chamber and President Putin before it becomes law.
If passed, the new rules will not take effect until September 2016 but will give the government grounds to block sites that do not comply.
"The ultimate goal is to shut mouths, enforce censorship in the country and shape a situation where internet business would not be able to exist and function properly."
But introducing the bill to parliament, MP Vadim Dengin said "most Russians don't want their data to leave Russia for the United States, where it can be hacked and given to criminals".
"Our entire lives are stored over there," he said, adding that companies should build data centres within Russia.