House leaders are hoping to spend the week of June 9 passing a new war supplemental, but negotiations were far from over last week as to whether the Senate and White House will agree to pay for domestic spending items in the bill.

“I think we are where we were two weeks ago,” said a Democratic source close to negotiations on the bill.

That is, House leaders are pushing for a war-spending bill that includes offsets for a $52 billion GI program and possibly extended unemployment insurance benefits. The Senate, meanwhile, lacks the votes to offset any domestic spending in the bill and President Bush is threatening to veto anything other than a clean supplemental.

Democrats are running out of time to get some kind of bill to the president’s desk before June 15, when the Pentagon is set to start issuing furlough notices to Defense Department employees. Technically, the funding hole could be covered by shifting funds around, but delaying further may be politically harmful to Democrats.

“We need to get a package that can be signed,” said the Democratic source.

Last week, Blue Dog Democrats offered a proposal for guaranteeing their votes on the war-spending bill: include offsets — currently, a tax increase on the wealthy — for expanded veterans education benefits. Then, leave it up to the Senate to take the risk of being blamed for defeating veterans benefits because of a tax increase.

But “there’s a reluctance” among House Democratic leaders to pursue this strategy, said the source, since the House already sent the Senate a war-spending bill with the tax increase and it was rejected.

“We did it once. It failed,” said the source. “A lot of people are saying, what is the net gain of doing it a second time?”

Further complicating matters is that Senate Republicans may try to attach more items to the bill if it comes back to them with offsets for domestic spending.

For now, fiscally conservative Democrats feel strongly about paying for any domestic spending in the bill, and “they’re going to fight it until the end,” said the source. “The question is, what happens at the end?”

As for extending unemployment insurance benefits, a provision that was originally in the House-passed bill, it remains up in the air whether House leaders will push to keep it in the bill the second time around.

For now, it is not contained in the bill, but “everybody thinks that could change,” said the source.

New national unemployment figures only added fuel to the debate over keeping the benefits in the bill. Labor Department data released on Friday show that unemployment jumped to 5.5 percent in May, the single-largest surge in more than two decades.

Speaker Nancy Pelosi (D-Calif.) issued a follow-up statement vowing to pass legislation to help people who have lost their jobs, but did not mention the supplemental.

House Majority Leader Steny Hoyer (D-Md.) said Thursday that he would like to have the package on the floor “earlier rather than later” in the week.

Outside of negotiations on the supplemental, the House will have a relatively quiet week. Lawmakers are set to debate two reauthorization bills, one for Amtrak and one for NASA. Both are set for Rules Committee hearings on Monday and Tuesday, respectively.

A Hoyer spokeswoman said it is possible that the House may also hold a veto override on the farm bill, for the second time, but it depends on when the White House is planning to veto the bill. For now, the White House has not scheduled a date.

The House and Senate already overrode a farm bill veto last month, but a technical error resulted in one section of the bill accidentally being left out of the bill before Bush’s veto. Both chambers have since passed the bill again, in its entirety, and now await the president’s follow-up veto to override it again.