U.S. Senate Minority leader Mitch McConnell (R-KY), answers questions during the weekly Republican news conference on Capitol Hill in Washington, U.S., September, 13, 2022.
"I'll proudly support the legislation provided that nothing more than practical changes are made to its current form," McConnell said in a floor speech.
"This bill, as introduced, is the only chance to get an outcome and to actually make law," he added.
In addition to the Jan. 6 violence, 139 House Republicans and eight Senate Republicans including Cruz voted to challenge results in key states.
The House bill would require at least one-third of the members of the House and Senate to agree to challenge a state's certification.