Senate Democrats say it was only because Senate GOP leaders decided to cut a deal on government funding and disaster aid that they were able to avert a government shutdown at the end of this week.

Speaker John Boehner (R-Ohio) refused to sign off on an offer that Senate Majority Leader Harry Reid (D-Nev.) made to him over the weekend, only to have brinkmanship-weary Senate Republicans undercut the Speaker by striking a deal with Reid on their own, according to a Senate Democratic leadership aide.

The reluctant embrace Monday of the Democrats’ bill by Senate Republican leaders sealed the agreement that allowed the continuing resolution to pass the Senate, Congressional aides said. The Democratic measure passed the Senate on Monday night 79 to 12.

Under the Senate-passed CR, the Federal Emergency Management Agency’s disaster relief fund would receive $2.6 billion. That is an effective cut of $1 billion from the House bill, but it allowed Senate Democrats to eliminate a $1 billion offset on which the House had been insisting.

The House-passed bill would have provided $3.6 billion for FEMA. Of the $3.6 billion, about $1 billion was targeted for fiscal 2011 and would have been offset by cutting two existing programs designed to subsidize alternative energy technology. The offset was included in the bill as an effort to try to reduce the deficit, which House Republicans have made their mission this Congress.

When Reid decided to eliminate the offset and the extra $1 billion in disaster aid, he was relatively confident that Senate Republicans would back the Democratic proposal, the Democratic aide said. Democrats reasoned that Senate Minority Leader Mitch McConnell would have trouble keeping GOP Senators from states affected by disasters from voting for the Democratic bill. After all, the Kentucky Republican had 10 defections on a more expensive disaster bill just two weeks ago.

Also, the Senate on Friday had defeated the House bill, on a motion to table, 59 to 36. And in a vote Monday, a different compromise from Democrats did not win enough votes to cut off debate.

So, according to Democrats, McConnell was faced with either voting Monday on the final Reid proposal and it winning the day, or waiting until closer to the government shutdown deadline of this Friday to cut a similar deal.

A Senate Republican leadership aide, however, said House and Senate GOP leaders were working together, fully aware of the situation and in agreement on how to proceed. The aide dismissed the notion that there was any division between House and Senate Republican leaders.

“Given a choice between the Speaker and Senate Democrats, we know where our fortunes lie,” the aide said.

Republicans contend that Congressional leaders last week had been working toward an agreement that entailed enacting a House-passed CR. But Senate Democrats decided to change course, which caused House Democrats to withhold their votes for the CR, according to House and Senate GOP aides. Democratic opposition and the defections of 48 Republicans caused the original CR to fail in the House last week. A Democratic aide said that there was never an agreement for the Senate to take the House measure.

Adding to the drama, both Senate Democrats and Republicans were unenthusiastic warriors. They had been scheduled to be on recess and working in their states this week, but they were engaged in the unexpected government shutdown fight, which also likely guided leaders’ decisions to accelerate a deal, aides said.

Republican aides said that the final compromise was the most obvious course of action after the White House declared that FEMA did not need the extra $1 billion in funding for the remainder of fiscal 2011, which ends this Friday.

The White House Office of Management and Budget sent a letter dated Monday, but publicly released today, to Reid assuring him that FEMA would remain solvent through this week.

House Republicans contend that that was what ultimately allowed Senate Democrats to push their proposal. Without the need for FEMA funding in fiscal 2011, there would be no need for offsets, which was at the heart of the dispute.

“The White House gave them an escape hatch,” a House GOP aide said.

House Republicans believe that Senate Democrats backed themselves into a corner after seeing an opportunity to fight the offset after House leaders lost the initial vote on their chamber’s CR.

“They got greedy and tried to play politics with disaster aid,” a House Republican aide said. “They thought they had us over a barrel.”

The House initially failed to pass the measure last Wednesday. House Democrats whipped against the bill and kept all but six of their Members from voting for the CR, a move that House GOP leaders said reneged on an earlier promise to deliver votes for the measure.

House leaders then held another vote Thursday after twisting arms to get the votes to pass the measure. After that, House leaders sent their Members home for a scheduled recess but also pressured Senate Democrats to accept their bill.

The shutdown fight was the third this year, and Sen. Mary Landrieu (D-La.) speculated that Senate Republicans are growing tired of their House counterparts.

“I think there is a weariness — and they may not admit it — but I sense there is a weariness among Senate Republicans about the operations of that House,” the Louisiana Democrat said. “And the public is growing tired of the tea party agenda that continues to put us on the brink of shutdown.”

House Republican leaders are expected to try to pass a short-term CR that expires Oct. 4 by unanimous consent this Thursday. The short-term CR, which was passed by the Senate on a voice vote Monday night, is designed to give House leaders a few days to vote on the Nov. 18 CR deal. If the short-term CR is passed, the House must then act by Tuesday to pass the Nov. 18 CR.

Steven T. Dennis contributed to this report.