Top Russian officials have expressed unease about the proposal, an apparent indication that the Kremlin opposes the move.
The lower house of parliament, the State Duma, on Wednesday approved the second reading of a bill responding to the U.S. legislation. The proposed adoption ban was not part of the original bill, but was added as an amendment.
To become law, the measure has to pass a third reading in the Duma, then clear the upper house before going to President Vladimir Putin for his signature.
Putin has not spoken publicly about the measure, but his spokesman Dmitry Peskov indicated Wednesday the Russian leader regards it as excessive. Peskov told the Interfax news agency that, although Putin understands the emotions that prompted the move, "the executive powers are taking a more restrained line."
Foreign Minister Sergey Lavrov and Russia's education minister both spoke out against the amendment and the speaker of the upper house, close Putin ally Valentina Matvienko, cautioned that the lower house was being guided by emotions rather than good sense.
But the overwhelming 388-15-1 vote in favor of the amendment Wednesday included lawmakers from the pro-Putin United Russia party. It was not clear if the opposition from top officials could bring about a reversal.
Many Russians have long resented the adoption of Russian children by Americans, sensitive to the implication that Russians are hard-hearted or economically unable to take care of their own. The resentment is fanned by cases of abuse or deaths of Russian children adopted by Americans.
Underlining the anger, the bill is named after Dima Yakovlev, a Russian-born toddler who died in the United States after his adoptive father left him in an car in the broiling heat for several hours. The father later was found not guilty of involuntary manslaughter.
The measure is in response to a new U.S. law, signed last week by President Barack Obama, which calls for sanctions against Russians assessed as human-rights violators.
That law stems from the case of Sergei Magnitsky, a Russian lawyer who was arrested by the Russian officials he accused of a $230 million tax fraud. He was repeatedly denied medical treatment and in 2009 died in jail. Russian rights groups have accused the Kremlin of failing to prosecute those responsible.
The amended passed by Duma on Wednesday also says any country that passes legislation similar to the Magnitsky Act also will be subject to an adoption ban.