But it's all very fluid. Republicans may be fine with a short-term debt limit extension, but then continue to insist on changes to ObamaCare as the price for reopening the government. Entitlement reforms will likely be another GOP demand — all of these could make it hard to find agreement with Democrats.

Still, the meeting is a sign that the two sides may finally start talking, on the 10th day of the shutdown and one week from the date at which Treasury says a debt ceiling solution is needed.

Obama already met with House Democrats on Wednesday, and will meet with Senate Democrats on Thursday.

Because of these meetings, Thursday will be relatively quiet on the floor. Republicans plan to take up one more "mini" spending resolution — H.J.Res. 79, the Border Safety & Security Act. This resolution would fund border security programs at the Department of Homeland Security.

The Senate starts at 10:30 a.m., and it will break at 1 p.m. for the Democrats' meeting with Obama.

The Senate is hoping to set up procedural votes on S. 1569 in the next few days; that bill would extend the debt ceiling through the end of 2014, allowing the government to keep borrowing above the current limit.