Moscow has denied it worked to influence the election in Trump's favour, and he has denied his campaign colluded.
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 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.
House Speaker Paul Ryan called the bill "one of the most expansive sanctions packages in history".
Representative Ed Royce, the Republican chairman of the House Foreign Affairs Committee, said the three countries "are threatening vital US interests and destabilising their neighbours. It is well past time that we forcefully respond".
- Reuters, AAP