Under intense pressure to take action on gun safety in the wake of two weekend massacres, Senator Mitch McConnell, the majority leader, expressed a new willingness Thursday to consider a measure expanding background checks for all gun purchasers, saying it will be “front and center” in a coming Senate debate on how to respond to gun violence.

“There is a lot of support for that,” he said.

Mr. McConnell, who has strongly opposed background checks in the past, made his remarks in an interview with a Kentucky radio host, Terry Meiners of WHAS in Louisville. While he did not support a bill requiring background checks, his remarks appeared to underscore the possibility of a shift in the politics of Washington’s divisive gun debate.