The proposed legislation could come to a vote as soon as this week.
Igor Sechin's comments follow similar claims from Putin himself.
Russian lawmakers deny allegations that Russia was behind the hacking of the Democratic Party during the election campaign.
The Treasury Department said U.S. companies could make limited transactions with the FSB.
Putin's spokesman said the leaders could meet before the G20 summit in July.
"We should not delude ourselves," Medvedev said.
Shuvalov has told ministers to assume sanctions will be lifted by the end of 2017.
Evidence seems to contradict the secretary of state nominee and former Exxon Mobil CEO's claims at confirmation hearing.
Foreign Minister Sergei Lavrov had proposed to Putin that Russia expel 35 U.S. diplomats.
Russia doubts the long-term effectiveness of new American measures to limit diplomatic ties, given that Obama is stepping down from office soon, a spokesperson for Putin said.
U.S. intelligence agencies have concluded that Russia tried to influence the Nov. 8 election by hacking individuals and institutions.
The CEO of the under-sanctions Sberbank hopes for new treatment from Trump.
Russians believe prices are rising but the Kremlin has vowed to maintain a ban on EU and U.S. imports "as long as possible."
The Russian president said the ban is helping Russian business.
The Foreign Ministry spokeswoman lashed out at a parliamentary report.
France's right-wing parties are looking favorably towards Russia.
The move will create 87,000 new jobs for Russians, according to the president.
Not only is Russia's president, Vladimir Putin, still standing, but the Russian economy, against most expectations, is actually recovering.
While the Russian health service stockpiles life-saving drugs, the president, MPs and their families may get them for free.
The EU "cannot and will not" drop sanctions before Russia fulfils ceasefire commitments, says Merkel.
British foreign minister says no plans to supply Kiev with weapons but 'not ruling anything out'.
The Russian economy has suffered severely due to the sanctions imposed by the West.