Sens. Mike Lee, R-Utah, and Rand Paul, R-Ky., voted against the Russian Federation sanctions package.
House leadership has not yet decided when or if it will take up the bill.
In the House, the Republican speaker, Paul Ryan, has signaled backing for the measures, although his party's support for the measure may not be as strong as in the Senate. The final Senate vote on the overall bill is expected in the coming days.
The measure also asserts a role for Congress if the White House opts to ease any sanctions against Moscow. The recent actions by U.S. Senate strongly suggest that.
Trump has repeatedly and openly doubted the veracity of the assessment.
Sanctions are authorized on Russia's mining and shipping sectors as well, and the government is required to study the USA economy's exposure to Russian state-owned enterprises.
Secretary of State Rex Tillerson said this week that he was wary of Congress taking actions that could interfere with the administration's efforts to improve relations with Russian Federation.
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But the Senate moved forward anyway.
Trump has said several times that he hoped to work with Russian Federation on fighting terrorism around the world, specifically the Islamic State group that has taken hold in Iraq and Syria.
Putin said that Russian Federation would be forced to make changes because of the sanctions, but they wouldn't lead to a "collapse".
The package would impose new restrictions on Russian actors linked to human-rights abuses, arms sales to the Syrian government or malicious cyberattacks on behalf of the Russian government.
- Allow broad new sanctions on key sectors of Russia's economy, including mining, metals, shipping and railways.
"This bill will impose sanctions on Iran for its non-nuclear violations", Cardin said. "I am strongly supportive of the sanctions on Russian Federation included in this bill", Sanders said in a statement.
"But I believe that these new sanctions could endanger the very important nuclear agreement that was signed between the United States, its partners and Iran in 2015", Sanders added.
In another challenge to Trump's foreign policy, the measure also "reaffirm the strategic importance" of North Atlantic Treaty Organisation.