The effort is led by Senate Foreign Relations Committee Chairman Robert Menendez.
The measure comes as the Biden administration tries to suppress defections on a competing effort targeting Russia, which will be put to vote in the Senate this week, the report noted.
The legislation would entail sweeping sanctions on top Russian military brass and government officials, including President Vladimir Putin, as well as key banking institutions.
It would also hit companies in Russia that offer secure messaging systems such as SWIFT. Pledging more security assistance to Ukraine, the bill calls on the United States to "consider all available and appropriate measures" to ensure Nord Stream 2 is not put into operation.
The White House supports the bill, a spokesperson for the National Security Council told The Washington Post, noting that other measures, such as the one being pushed by Senator Ted Cruz, will "not counter further Russian aggression or protect Ukraine."
"Instead, it will undermine our efforts to deter Russia and remove leverage the United States and our allies and partners possess at this moment all to score political points at home," the spokesperson said on the condition of anonymity. "And it would come at a moment where we need to be closely united with our European partners, including Germany. It makes no sense."