Senate Majority Leader Mitch McConnell is negotiating a way to pass more stringent sanctions. (Oliver Contreras/For The Washington Post)

A group of leading senators, including Majority Leader Mitch McConnell (R-Ky.), is negotiating a way to pass more stringent sanctions against Russia in the coming week by piggybacking on an upcoming a measure cracking down on ballistic missile tests in Iran.

The talks, which involve the heads of at least the Senate Banking and Foreign Relations committees, plus Senate leaders and a handful of Congress’ most outspoken Russia critics, are geared toward attaching Russia sanctions by amendment to an Iran sanctions bill the Senate took up Wednesday — just as intelligence and Justice Department officials head to Capitol Hill to testify about alleged Russian meddling in the presidential election.

Daniel Coats, the director of national intelligence, and Adm. Mike Rogers, the director of the National Security Agency, appeared before the Senate Intelligence Committee on Wednesday, and former FBI director James B. Comey is expected to testify Thursday.

“We anticipate that amendments addressing Russia sanctions are likely to be offered,” McConnell said on the Senate floor Wednesday morning, noting that “I support that effort” from the Foreign Relations and Banking Committee chairmen and ranking members “to work toward bipartisan agreement.”

Senate Minority Leader Charles E. Schumer (D-N.Y.) was even more insistent, all but leveraging his party’s support for the otherwise popular Iran sanctions bill on whether punitive measures against Russia could be incorporated.

“It will be very difficult to gather Democratic support for final passage of this bill until we deal with Russia sanctions,” Schumer said Wednesday.

The exact substance of the Russia sanctions senators hope to attach to the Iran bill is not yet clear, but according to senior Senate aides, talks have focused on the substance offered by a set of bills already on offer, addressing everything from Russia’s aggressive activities in Ukraine and Syria to allegations that Russian hackers tried to swing an American election.

One of those bills is a recent measure, from Senate Banking Committee Chairman Mike Crapo (R-Idaho) and ranking Democrat Sherrod Brown (Ohio), to codify into law existing sanctions against Russia that the Obama administration imposed in response to Russian aggression in Ukraine in 2014. Their bill adds to those existing sanctions new sectoral measures against Russia’s mining, metals and railways; individual sanctions against Russian hackers and corruption; and tools to better track illicit Russian financing, as well as oligarchs’ holdings in the United States.

Another bill, proposed this year by Sens. Benjamin L. Cardin (D-Md.), John McCain (R-Ariz.) and a bipartisan group of senators, would codify existing sanctions while stiffening restrictions on the Russian defense, intelligence and energy sectors, as well as anyone providing material support to people posing a cyberthreat. A third bill from Sens. Lindsey O. Graham (R-S.C.), Cardin and others, also from this year, would give Congress the chance to veto any presidential decision to ease up on sanctions against Russia.

All chief authors of those bills have been involved in the various discussions with Senate leaders McConnell and Charles E. Schumer (D-N.Y.) to insert Russia sanctions into the Iran bill, according to various aides.

Which elements of those Russia sanctions bills make the cut to be included as a potential amendment to the Iran sanctions legislation has yet to be determined. But the coalescing of forces around some sort of action on Russia sanctions — and soon — is a marked shift for Congress, which has to date refrained from taking legislation action to force the president’s hand in dealing with a country many in the national security community consider America’s No. 1 antagonist and adversary.

Some Republican leaders have sought to create some space for the Trump administration to operate, pointing to that fact despite initial fears Trump’s administration would take early steps to roll back Russia sanctions, the restrictive measures the Obama administration introduced have stayed in place. Early last month, Senate Foreign Relations Committee Chairman Bob Corker (R-Tenn.) insisted the Senate Foreign Relations Committee would wait until the Senate Intelligence Committee completed its probe of alleged Russian election meddling and possible collusion with campaigns before attempting to pass any sanctions.

President Trump has pushed back against the consensus of the intelligence community that Russia engaged in hacking and the politicized dissemination of information, including false information, during the campaign to sway the election, potentially in Trump’s favor. According to recent reports, Trump also pressured Comey and other senior intelligence community officials to use their influence to shut down the FBI’s investigation into potential ties between Trump associates and campaign officials and the Kremlin.

But as allegations mount of more and worse Russian interference in the 2016 election, the pressure to respond with punitive action has been increasing on both sides of the Senate, with even the Republican leader indicating a willingness to take up the issue on the floor. Late last month, Corker indicated that if Secretary of State Rex Tillerson could not show in early June that progress was being made with Russia with respect to the war in Syria, he would green-light new Russia sanctions in the Senate, and one that included measures to censure Russia over its alleged election meddling.

On Tuesday, after a check-in call with Tillerson Monday night, Corker hinted that a Russia sanctions measure would likely be coming soon — but he cautioned against making any assumptions about its contents, or presuming that any already-filed bill would be the model for it.

If senators are to use the Iran sanctions bill as a vehicle for Russia sanctions, they will have to make those determinations soon, as the Senate takes an opening procedural vote on the legislation Wednesday, setting it up for passage likely as soon as early next week. The Iran sanctions bill coming before the Senate seeks to punish Tehran for a spate of recent ballistic missile tests, as well as the activities of the Islamic Revolutionary Guard Corps, a group the Trump administration is considering labeling a terrorist organization.

Corker, who chairs one of the two Senate committees with chief jurisdiction over sanctions, has not attached his name to any pending Russia sanctions bill. Crapo, who chairs Banking, the other committee of jurisdiction, is the lead co-author on that committee’s legislation, filed last week.