Obama this year called on Congress to give him this authority in the context of an agreement aimed at avoiding some of the consequences of the "fiscal cliff," but many Republicans have rejected this idea as one that cedes too much power to the president.

Reid's bill, however, does not fully vest the power to raise the debt ceiling with the executive branch. Instead, it allows these increases to happen either through the usual congressional process or through an executive order issued by the president.

Under Reid's bill, these executive orders would be reviewable under the CRA. That means Congress would have the option of passing a resolution disapproving of these executive orders.

Assuming the president supports his own executive order and would veto a resolution of disapproval, Congress would effectively need to approve these resolutions by a two-thirds majority vote to override that veto.

On Thursday, Reid sought a unanimous consent agreement to take up his bill. But McConnell (R-Ky.) said he would only support consideration if it required a 60-vote majority, which would ensure its defeat in the Senate.

After his attempt failed, Reid vowed to continue his efforts to bring up the bill.