WASHINGTON — Senator Mitch McConnell endorsed a bill on Tuesday to overhaul how Congress counts electoral votes to confirm the results of a presidential election, significantly enhancing the prospects of enacting the most substantial legislative response yet to the Jan. 6, 2021, attack on the Capitol.
The support from Mr. McConnell, the Kentucky Republican and minority leader, represented a substantial break with his party in the House, where all but nine Republicans opposed a similar measure that passed last week. It came as the Senate Rules Committee delivered an overwhelming bipartisan vote to send the legislation to the floor.
“The substance of this bill is common sense,” said Mr. McConnell, a member of the Rules Committee, about the legislation negotiated in recent months by a bipartisan group led by Senators Susan Collins, Republican of Maine, and Joe Manchin III, Democrat of West Virginia.
One crucial piece of the measure spells out that the role of the vice president, who presides over the counting of the electoral votes as the president of the Senate, is strictly ceremonial. That provision is a direct response to the failed effort by President Donald J. Trump and his allies to persuade Vice President Mike Pence to reject presidential ballots cast in favor of Joseph R. Biden Jr. as part of a scheme to invalidate his victory.
The legislation also seeks to prevent state officials from submitting electoral votes that do not align with the popular vote in a state, another answer to Mr. Trump’s election subversion attempt, which included a bid to have allies submit slates of pro-Trump electors in states won by Mr. Biden. It would substantially increase the threshold for Congress to consider an objection to electoral votes, requiring that at least one-fifth of each chamber sign on to such challenges, which currently need only one senator and one House member.
“Right now, just two people out of 535 members can object and slow down and gum up the counting,” said Senator Amy Klobuchar, Democrat of Minnesota and the chair of the panel. She said the legislation presented “an opportunity to take strong bipartisan action to protect the cornerstone of our democracy: the peaceful transfer of power.”
The Rules Committee incorporated some changes sought by election watchdogs, and Mr. McConnell warned that he would not back any final version if the authors went beyond narrow consensus changes. He also said he would not support the bill approved last week in the House. That legislation raises the threshold for objections even higher, to one-third of both chambers, and also includes provisions that Senate Republicans fear could lead to more court fights over the results — an outcome they are eager to avoid.
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“It is clear that only a bipartisan compromise originating in the Senate can actually become law,” Mr. McConnell said. “One party going it alone will be a nonstarter and, in my view, the House bill is a nonstarter. We have one shot to get this right.”
Senator Ted Cruz, Republican of Texas and one of the senators who lodged objections to the electoral count in 2021, was the sole member of the Rules Committee to oppose the legislation.
“This bill is all about Donald J. Trump, and in our lifetimes, no one has driven Democrats in this body more out of their minds than President Trump,” said Mr. Cruz. He said the legislation raised serious constitutional questions and was an attempt by Democrats to gain control over elections that should remain the responsibility of the states.
“I don’t believe senators from this side of the aisle should be supporting a bill that enhances the federalization of elections and reduces the ability of Congress to respond to the very serious problem of voter fraud,” he said.
Senator Chuck Schumer, the New York Democrat and majority leader, also made clear that he was behind the legislation. A spokesman said that Mr. Schumer “looks forward to continuing to have bipartisan, bicameral discussions about the best way to ensure Electoral Count Act reform legislation is signed into law soon.”
Backers of the bill say it must be approved this year and would have no chance if Republicans claimed control of the House in the upcoming midterms.