Outlets such as Voice of America and Radio Free Europe/Radio Liberty, which receive funding from the US Congress, would be forced to register as foreign agents.
Its passing follows RT registering with the US Justice Department following claims from US intelligence agencies that the television channel served as a Kremlin tool to meddle in the 2016 presidential election.
The third reading is seen as a formality, after which the amendments will have to be passed by the Senate and then be signed into law by President Vladimir Putin.
US investigators are reportedly looking into whether Russian government-funded outlets such as RT and Sputnik were part of Russia's influence campaign aimed at the 2016 presidential election.
Russian Federation denies it meddled in the 2016 election, and RT has accused the Justice Department of impinging on free speech.
Legislator Leonid Levin said the bill will provide a tool for the Justice Ministry to designate worldwide media outlets as foreign agents.
Putin's spokesman Dmitry Peskov said the legislation will provide the necessary framework for the government to retaliate to any foreign action against the Russian media.
Senior Russian lawmaker Andrei Isayev said he hoped German media would not be affected by the new regulations.
According to the bill, a media organisation classified as a foreign agent will be subject to the same requirements that are applied to foreign-funded non-governmental organisations under a 2012 law.
A Russian law adopted in 2012 forces NGOs that have worldwide funding and whose activities are deemed "political" to undergo intensive scrutiny of their finances and staffing and label themselves as "foreign agents" on paperwork and statements.
Critics of the law have said the definition of political activity is so loose that it could be used against nearly any non-governmental organization.
Amnesty International criticised the new bill as an attack on media freedom. Yahoo News has reported that the Federal Bureau of Investigation has interviewed a former Sputnik correspondent about his work at the website.
The report said that RT "conducts strategic messaging for [the] Russian government" and "seeks to influence politics, [and] fuel discontent in the U.S".