Senate Majority Leader Harry Reid signaled Tuesday that a deal is near to end the stalemate over funding the Federal Aviation Administration, which partially shut down last month, leaving thousands of civilian employees and construction workers on furlough.

Reid indicated that the end of the impasse could come with his accepting a funding bill from the Republican-controlled House that includes a provision eliminating millions in subsidies for rural airline services -- a provision that Senate Democrats had opposed.

"Sometimes you have to be reasonable, as we learned with this big deal we've just done," said Reid, D-Nev. "Sometimes you just have to step back and do what's best for the country and not be bound by your personal issues, and I'm willing to give that up, and I hope the other side will do the same."

With everybody's energy devoted over the last couple weeks to solving the debt-ceiling crisis, little progress has been made toward resolving the FAA dispute. And there was concern earlier that an agreement wouldn't be possible until Labor Day since lawmakers were expected to be out of Washington for the rest of the month

"Our position is the Senate has had our bill for weeks and they need to pass it," a senior House GOP source told Fox News. "There is no chance we'd take something different up in pro forma, we've told the Senate that."

An estimated 4,000 FAA workers have been furloughed, tens of thousands of airport construction workers are in limbo and at least $30 million a day in airline taxes are not being collected because of Republican objections to provisions that they called wasteful and a Democratic reach-out to union supporters.

In a notice posted on the FAA employee website and obtained by Fox News Radio, Air Traffic Organization Chief Operating Officer David Grizzle raises safety concerns as the furlough drags on. But an FAA official told Fox New Radio that the note references construction sites ATC facilities that were left in a haphazard state due to the abrupt nature of the shutdown. The official maintained there is no safety issue with the traveling public.

In a Rose Garden speech delivered after the Senate sent him legislation to extend the nation's borrowing limit, President Obama urged Congress to end the standoff .

"It's another Washington-inflicted wound on America and Congress needs to break that impasse now," Obama said.

But members of Congress continued to blame each other for the impasse.

House Minority Whip Steny Hoyer blasted Republicans for leaving town without resolving the FAA dispute, calling it "the politics of confrontation."

On Monday, Transportation Secretary Ray LaHood said, "Congress should not leave on its summer vacation until they finish an FAA bill that sends back to their desks and their offices 4,000 FAA employees and sends back to the workplace 70,000 construction workers,"

The areas of disagreement center on a House Republican provision added to a long-term FAA funding bill making it more difficult for airline workers to unionize. That bill then stalled, and lawmakers turned to a short-term extension -- the kind of bill the FAA has been operating under since 2007.

But that effort stalled when Republicans added a separate provision stripping $16.5 million in subsidies for rural airline service.

The Senate had been working on a separate proposal. Aides to Sen. Jay Rockefeller, D-W.Va., said in a memo that they're looking to bring a "clean" short-term funding bill to the floor Tuesday, urging the House to support it. Claiming Democrats have since called for FAA funding cuts beyond what Republicans proposed in a long-term package, the memo said Republicans would be "completely to blame" for the FAA's woes if they object to the "clean" short-term bill meant to last through the end of September.

Rep. Nick Rahall, D-W.Va., accused Republicans of taking a "my way or the runway" approach, claiming they were scoring "political points for Tea Party extremists" at the expense of U.S. jobs and lost taxpayer revenue.

But Rep. John Mica, R-Fla., chairman of the House Transportation Committee, accused the Senate of choosing "political pork over American workers."

He said the change to the so-called Essential Air Service would merely cut funding, some of which he described as "exorbitant," for airports within 90 miles of larger airports.

"Apparently, protecting outrageous airline ticket subsidies is more important than putting 4,000 furloughed FAA employees and thousands of airport construction workers across the country back to work," he said.

The controversial labor provision holding up the long-term bill would overturn a decision by the National Mediation Board allowing airline and railroad employees to form a union by simple-majority vote. Republicans, who have long been concerned about union intimidation in these votes, want to keep the former rule treating a non-vote as a "no" vote.

In the meantime, an inter-agency memo put out Monday by the FAA office of the Chief Counsel and obtained by Fox News Radio suggests the agency was preparing for the long haul. The memo advises FAA employees on such matters as applying for unemployment compensation and rules on moonlighting.

Fox News' Molly Henneberg and Fox Radio's Mike Majchrowtiz contributed to this report.