WASHINGTON — For two decades, through Columbine, Virginia Tech, Newtown, Orlando, Las Vegas and the rest, mass shootings have provoked only scant action in Congress. Now, after horrific back-to-back massacres this weekend, people in both parties agree that one man could change that: Senator Mitch McConnell.

As President Trump urged national unity in the face of “racist hate” without endorsing broad gun control measures, Democrats and a handful of Republicans called on Monday for Mr. McConnell, the majority leader, to bring up legislation to require gun buyers — including those on the internet and at gun shows — to go through background checks.

Mr. McConnell, nursing a fractured shoulder from a weekend fall at home in Kentucky, made no commitments. After consulting with advisers and fellow Republicans, he issued a statement Monday evening saying that he had asked three top committee chairmen “to engage in bipartisan discussions” about how to address gun violence “without infringing on Americans’ constitutional rights.”

“Senate Republicans are prepared to do our part,” he said.

But just what “our part” is, was unclear. One of Mr. McConnell’s longtime advisers, J. Scott Jennings, said Monday that Mr. McConnell was positioned to lead a process in Congress that could “achieve a solution that, at a minimum, comforts the nation.”