The House of Representatives passed the legislation by a margin of 419-3, clearing the way for the Senate to approve the sanctions created to punish Moscow over alleged meddling in the United States presidential election.
"These three regimes in different parts of the world are threatening vital USA interests, and they are destabilizing their neighbours", House Foreign Affairs Committee Chairman Ed Royce said on Tuesday.
The package includes sanctions that passed the House in May to choke off money flows to North Korea's missile and nuclear programs.
Kushner, who is also President Donald Trump's son-in-law, testified before the House and Senate intelligence committees this week over his contacts with Russian officials during the election campaign. If the US President vetoes the bill, Congress members have threatened to override this veto given the vast bipartisan majority behind the measure.
However, it also prevents the president from rolling back the sanctions without congressional approval.
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A European Union document prepared for the commissioners, laid out the EU's plans to seek "demonstrable reassurances" that the White House would not use the bill to target European Union interests.
The deal comes after Corker and McCarthy publicly disagreed on the future of the sanctions, raising questions about whether the bill would reach Trump's desk before he left for recess in August.
The bill still must be passed by the Senate, which is mired in debate over efforts to overhaul the USA healthcare system as politicians try to clear the decks to leave Washington for their summer recess.
Moscow had initially hoped that Trump would work to fix a relationship which has slumped to a post-Cold War low, but has watched with frustration as allegations that Moscow interfered with last year's USA presidential election and concerns over Trump associates' Russian Federation ties have killed off hopes of detente.
Oil and gas companies had also raised concerns about provisions that would have restricted interactions between American and Russian energy companies.
Vladimir Putin and oligarchs close to him are all targeted by the sanctions.
"We consider such a continuation of the rhetoric of sanctions counter-productive and harmful to the interests of both countries", it said.