The amendments to an existing information law are being promoted as a crackdown on child pornography, creating a federal register that would rule out websites carrying banned information and oblige site owners and providers to close down the sites.
The Russian-language version of Wikipedia went on strike the day earlier in protest at the bill. "Imagine a world without free knowledge," it said, blocking access to the site. The site was back up on Wednesday.
The bill has been scheduled for approval in second and third readings Wednesday after the initial reading last Friday, with next to no debate or public discussion.
Russia's biggest search engine Yandex said the bill's proposed methods to fight pornography "create room for possible misuse and raise questions from Internet users and company representatives."
"Such decisions cannot be taken hastily, the way it is happening now," said the statement signed by the company's chief editor Yelena Kolmanovskaya. The engine's slogan "Everything will be found" had the word 'everything' crossed out on the main page on Wednesday.
An expert on Russia's security services, Andrei Soldatov, said the bill would introduce a technical possibility of blocking foreign sites for the first time by forcing Internet providers to install special equipment.
"Clearly, it will be possible to use it not just against websites propagating pornography; the government will be able to use these instruments any way it wants," he wrote on his website Agentura.ru.
"The amendments can lead to introduction of censorship to Russian Internet," said Livejournal, a popular blogging platform frequently used by opposition leaders for communicating with their audience.
Opposition deputies on Wednesday decried what they said was a trend of introducing restrictive bills at short notice and ramming them through without any public discussion.
"We are turning the parliament into an secretarial office that carries out somebody else's wishes," opposition deputy Gennady Gudkov of the Just Russia party said at the hearings.