Senate Majority Leader Harry Reid (D-Nev.) warned Tuesday there is a chance that the government could shutter by the end of this month.

Criticizing House Republicans for the disaster relief provisions in their budget bill, Reid told reporters, "We're not going to cave on this."

Senate Minority Leader Mitch McConnell (R-Ky.) on Tuesday confidently predicted the two sides will come together by Thursday night: "There wil not be a government shutdown."

Reid, however, said, "I'm not that sure" there won't be a shutdown. He added, "I am not as certain as McConnell."

At issue is increased funding for the Federal Emergency Management Agency (FEMA) to help a number of states recover from various natural disasters. The government funding bill, known as a continuing resolution, that the House will take up on Wednesday contains $3.65 billion for FEMA, but the Senate has previously approved stand-alone legislation that would provide $6.9 billion in relief funds. Reid announced Tuesday he would be amending the House bill to include that increased funding.

Tuesday saw each party accusing the other of playing games with legislation to keep the government funded through Sept. 30. House Majority Leader Eric Cantor (R-Va.) accused Reid earlier in the day of playing a "political game," asserting that the Senate Democrat would be solely to blame if a shutdown occurred.

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Meanwhile, Sen. Mary Landrieu (D-La.), who appeared alongside Reid at a press conference, insisted the House GOP approach was "inadequate" and "wholly unprecedented" for requiring immediate offsets to pay for the disaster funding.

"We will eventually pay for it, but we can argue about that later," she said.

The two Democrats ramped up pressure on the 10 Republican senators, almost all in disaster-affected areas, who previously voted for the stand-alone bill with higher funding. If those Republicans opposed his version of the bill, it would send a "very, very sad message to home that because of partisanship they're going to back off on what's needed," Reid said.

Reid said the deadline for getting a deal on a short-term continuing resolution is Sept 30, the end of the fiscal year. He also indicated that Congress could cancel its scheduled recess for the last week of September if a spending deal is not reached.

"If they want to stay into next week, that's fine we can do that," he said. "We can work all next week."

Peter Schroeder contributed to this story.