McConnell, speaking from the floor, signaled that he thought the commission would be redundant since the Justice Department and Senate committees are months into their own investigations.
"There is, has been and there will continue to be no shortage of robust investigations by two separate branches of the federal government," McConnell said.
"It's not at all clear what new facts or additional investigation yet another commission could actually lay on top of existing efforts by law enforcement and Congress," he added.
McConnell, echoing concerns from some of his members, also appeared to take issue with potential limits on the investigation, including on its scope. McConnell, backed up by McCarthy, has argued that it should stretch into other violence including protests, some of which turned violent, in the wake of police-involved deaths.