Story highlights The bill has already been rejected by the President and Republican leadership

Advocates hope Democratic unity can put negotiations back into play

Washington (CNN) As votes for government funding look perilous in Congress, bipartisan senators behind an immigration deal are furiously working behind the scenes to build support for their bill, hoping it could be in play to avert a shutdown.

The group, an offshoot of the so-called Gang of Six, is "practically sprinting" to get the bill officially introduced, one congressional aide said, and are working to add as many Republican supporters as possible. When the bill was unveiled on Wednesday night, it had picked up four Republicans in addition to the three that worked to develop it.

If all 49 Democrats support the bill, only four more Republicans would be needed to clear the 60 votes required to advance legislation in the Senate.

Even though the bill has already been rejected by the President and Republican leadership, the calculus is that with a standoff on government funding, Republicans will be pressed on why they walked away from a bipartisan deal with votes to pass it when the shutdown blame game begins.

Illinois Democratic Sen. Dick Durbin, one of the lead authors of the bill, met on Thursday morning with the House Problem Solvers Caucus, a bipartisan group of lawmakers who have sought a centrist deal on DACA as well, his office said. Building House support could answer White House chief of staff John Kelly's criticism that the bill didn't have backing from both sides of Capitol Hill when it was brought to the President last week.

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