Advocates for congressional action in both parties are not giving up.

“Leader McConnell would just like the issue to go away,” Senator Chuck Schumer of New York, the Democratic leader, said this week in an interview. “We’re not going to let that happen.”

Mr. Schumer and others point to continued warnings from the American law enforcement and intelligence officials monitoring Russia and other foreign adversaries — including Mr. Mueller — who say the threat against the American political process may only be growing. Adding to concerns are reports that Mr. Trump’s sensitivity around Russia might be hindering the executive branch’s response and surveys that show more than a dozen states still do not have auditable paper-ballot backup systems. Dozens more say they lack the funds to replace aging and potentially vulnerable election technology.

“We have 8,000 different election jurisdictions, and the idea that all of them are going to have the resources, the knowledge, the skills and the ability to independently safeguard our system against foreign powers is just not realistic,” said Lawrence Norden, who has surveyed state election officials for the Brennan Center for Justice. “We need somebody to be leading on this. In some cases, individual states are. But given the threat, the idea that Congress is not going to have a role here for ideological purity, to me is insane.”

Democrats in the House majority are readying hearings and votes on election-related measures to try to force Mr. McConnell’s hand, party aides said. The House Intelligence Committee, one of the panels involved, announced Friday that it would hold a series of hearings on the Russian counterintelligence threat detailed by Mr. Mueller, including the need for additional legislation.

The votes will include one on the Election Security Act, a sweeping but partisan bill that proposes spending $1 billion in grants to state and local officials for replacing voting machines, hiring information technology staff and funding other security measures. But the bill also mandates the president develop a national strategy to fend off influence operations and disinformation campaigns like the ones Russia executed and promulgates new standards for vendors of election technology.

The House may also consider narrower, bipartisan bills Democrats hope could attract more Republican support, like the Honest Ads Act, which would force Facebook, Google and other internet companies to disclose who is purchasing political advertising, and a bill focused solely on getting states to adopt the use of backup paper ballots.

Daniel Savickas, who lobbies Congress on election-related issues for the conservative FreedomWorks, blasted the Senate majority leader for letting legislation languish: “Unfortunately, all Senator McConnell wants to do is judges these days,” he said.