The legislation, which parliament's lower house, the State Duma, unanimously backed in a fast-tracked vote last week, would allow Moscow to target foreign media outlets in a similar way it has gone for NGOs that receive international funding.
Many NGOs have closed in response to the intense scrutiny.
Under the new legislation, US and other foreign media would have to present themselves as such on all paperwork and submit to intensive scrutiny of staffing and financing.
The Kremlin's council for human rights had recommended the Senate reject the bill, saying it was vaguely worded and difficult to reasonably enforce.
"Any foreign media could be labelled a foreign agent quite arbitrarily by the justice ministry of the Russian Federation," the council said in a statement on Monday.
Russia's justice ministry said last week it had already contacted the US-funded Voice of America and Radio Free Europe/Radio Liberty to inform them they might have to start labelling themselves as "foreign agents".
Washington considers RT a propaganda arm of the Kremlin and told it to register its American operation under the Foreign Agents Registration Act, which is aimed at lobbyists and lawyers representing foreign political interests.
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