On a second try, the Senate voted Tuesday to take up a House-passed bill that Senate Majority Leader Harry Reid (D-Nev.) wants to use as a vehicle for about $7 billion in emergency disaster relief.

The Senate voted 61-38 to proceed to the bill, with eight Republicans voting with Democrats, including Sen. Pat Toomey of Pennsylvania, whose state has suffered major flooding.

Other GOP backers included Senators from states that have been hit hard by recent storms, such as David Vitter of Louisiana, Roy Blunt of Missouri and John Hoeven of North Dakota.

Action on the measure came after Reid tried and failed Monday to win the 60 votes needed. In that vote, the motion to proceed received 53 votes, but a number of Senators who would likely have backed Reid did not vote.

Senate passage would set up a standoff with House Republican leaders who have said they want to provide disaster aid in the continuing resolution that will be needed to give Congress additional time to pass the annual spending bills for fiscal 2012. The fiscal year ends Sept. 30.

Democrats have also been critical of House Majority Leader Eric Cantor’s proposal to offset the disaster aid with spending cuts elsewhere in the budget. The Virginia Republican has since said the disaster funding will be able to fit within flexible spending caps set by the August debt ceiling deal.

Before Tuesday’s vote, Senate Majority Whip Dick Durbin predicted that the measure would pass.

The Illinois Democrat also said that although he would like to see the cost offset, he does not want the emergency funds to become the center of a fight over offsets that would delay delivery of the money.

“We have to pay for these disasters across America,” Durbin said. “Historically, we have taken care of those disasters and not subjected them to long-term political wrangling.”

Sen. Dean Heller also voted to take up the bill, just as he did Monday. The Republican said his state of Nevada has concerns about wildfires, and he wants to make sure his constituents get aid if there are any future disasters in his state.

“Obviously, I want it paid for,” Heller said. “I just want to make sure we maintain protection for the homeowners in Nevada.”

Some conservative Senators have urged leaders to allow a vote on offsetting the spending. Sen. Rand Paul (R-Ky.) has suggested offsetting the cost by cutting foreign aid but hasn’t said what specific programs should be cut.