"If the Ten Commandments were in the bill, the NRA would say that it had gun registry," he said.

The big-ticket items with the best chance of passage are the universal background checks bill and the gun trafficking bill, which could gain support from Republicans in the wake of the "fast and furious" incident. A ban on high-capacity magazines looks unlikely but not impossible. Also under discussion are ways to incentivize states to do a better job at making mental health records available.

Democrats will almost certainly need to overcome a filibuster -- which requires 60 votes -- to pass the bill through the Senate. There are now 55 senators who caucus with Democrats, including some who hail from relatively pro-gun states and may be skittish about backing the measure.

If Coburn and other NRA-backed Republicans decide to vote for a bill, it will increase pressure on the House Republican leadership to bring it to the floor. Even if the bill comes to the floor, it's far from clear that enough Republicans and pro-gun Democrats would be willing to vote for it to get it to the president's desk.

"We are working," Schumer told CBS News late Tuesday. "Guns are not an easy issue but we are working. That's where we are at."