The White House said the President had not yet decided whether he would sign the measure. Rejecting the bill -- which would potentially hamper his hopes of pursuing improved relations with Moscow -- would carry a risk that his veto could be overridden by lawmakers.
"While the President supports tough sanctions on North Korea, Iran and Russia, the White House is reviewing the House legislation and awaits a final legislative package for the President's desk," said spokeswoman Sarah Huckabee Sanders.
House members backed the bill, which also imposes sanctions on Iran and North Korea, by a near-unanimous margin of 419-3, with strong support from Trump's fellow Republicans as well as Democrats, despite objections from Trump, who wanted more control over the ability to impose sanctions.
The Republican-controlled Senate passed an earlier version of the bill with near-unanimous support. The House added the North Korea measures after becoming frustrated with the Senate's failure to advance a bill it passed in May.
The intense focus on Russia, involving several congressional probes and a separate investigation by a Justice Department-appointed special counsel, Robert Mueller, has overshadowed Trump's agenda.
The scrutiny has angered and frustrated the President, who calls the investigations a politically motivated witch hunt fuelled by Democrats who cannot accept his upset win in last November's election against Democratic candidate Hillary Clinton.
The Senate Judiciary Committee had been set to compel Trump's former campaign manager, Paul Manafort, to testify at a hearing today, but rescinded the subpoena late yesterday as negotiations over his participation continued.
Manafort has started turning over documents to the committee and is negotiating a date to be interviewed, the panel said in a statement.
The committee is looking at a June 2016 meeting in New York with a Russian lawyer organised by Trump's eldest son, Donald Trump Jnr. Trump's son-in-law, Jared Kushner, spent three hours with the House of Representatives intelligence panel, his second straight day on Capitol Hill answering questions about his contacts with Russians during the campaign.