WASHINGTON — The House on Tuesday approved a sweeping package of sanctions against Russia, clearing a key hurdle in Congress’s effort — long opposed by the Trump administration — to punish Moscow for its aggression toward its neighbors and its interference in last year’s American presidential election.

The 419-3 vote brings President Trump one step closer to a choice he has strained to avoid: whether to sign legislation embraced by Republicans in both the House and the Senate that undercuts his attempts to ratchet down tensions with Moscow, or to veto the bill even as Russia-related scandal consumes his administration.

The measure would sharply limit the president’s ability to lift or suspend sanctions. It also includes sanctions against Iran and North Korea, two countries the administration has been more eager to hold to account.

The legislation must be taken up by the Senate before being sent to the president’s desk. Senator Mitch McConnell of Kentucky, the majority leader, has not said whether that might happen before the August recess. The Senate supported a similar bill nearly unanimously last month, but it…