The senators said 14 states had used at least some machines in 2016 without a paper trail that could be audited. Five states used only those machines.

At the same time, they took pains to make clear that while it was the states’ responsibility to run their own elections, the federal government should play a significant role in monitoring threats and providing resources. Both were to blame for missteps in 2016, they said.

“We were all disappointed that states, the federal government, the Department of Homeland Security were not more on their game in advance of the 2016 elections,” Mr. Warner said. While the department had made important improvements since then, he added, it was still a work in progress.

The senators pointed to top state election officials, who have complained in recent weeks that the federal government has left them in the dark on specifics of the threat to voter databases, machines and other systems.

The Intelligence Committee is likely to try to illustrate those breakdowns and possible solutions on Wednesday during a hearing with Homeland Security and state election officials on attempted attacks by the Russians on state election systems in 2016 and the federal response.

The committee has also assembled a classified written report on election security. But the document has to be submitted to the intelligence agencies for review and declassification before it can be released publicly, which could take weeks or months.

More than a year into its investigation into Russia’s meddling, the Intelligence Committee has thus far offered little visibility into its work. Witnesses have come and gone in relative secret. Staff members and lawmakers have pored over thousands of pages of documents related to the Trump campaign, as well as sensitive government secrets.