“Unfortunately, Senator McConnell, self-described crepe hanger, has vowed to kill our bill as the president declares he sees no problem with foreign intervention in our election,” Speaker Nancy Pelosi of California said Thursday morning on the floor of the House. “The G.O.P. Senate and the White House are giving foreign countries the green light to attack our country. But the House will do our part to protect America.”

Mr. McConnell has showed little sign of budging. Colleagues say that he, like some other members of the Republican Party party, ideologically opposes the federal government wading further into election administration, which has traditionally been carried out by the states. And he has expressed confidence that a 2017 package of Russia sanctions and $380 million in grants Congress allocated to states last year, along with executive branch actions to step up deterrence and support state election authorities, have already set the country on the right path.

In a brief interview on Thursday, Mr. McConnell called the House bill a “nonstarter” in the Senate.

The bill, the Securing America’s Federal Elections Act, would set new federal standards for vendors and operators of voting systems that most security experts argue will significantly limit the risk of foreign intrusions. In addition to authorizing $600 million for Election Assistance Commission grants to update voting technology, the bill would require states to use backup paper ballots that can be verified by voters and election counters, institute postelection audits to search for possible irregularities and prohibit voting systems from connecting to the internet.

Many states have already adopted those changes, but experts who argue for federal assistance say not all local election authorities have the money, personnel or know-how to keep systems updated. The $380 million that Congress signed off on last year helped, they say, but it only began to address the need.

“The states are not responding with the alacrity that this situation demands,” said Susan Greenhalgh, a vice president for the National Election Defense Coalition, a nonpartisan group trying to unite the left and right around election security issues. “It is a national security crisis — nothing less — and that requires a federal response.”