WASHINGTON -- It may be too late to for a grand bargain to avert the so-called fiscal cliff before the nation starts to slide over it, but there is still time to ease some of the biggest problems -- as long as House Republican leaders are willing to go along.

Following a meeting of political leaders at the White House Friday, Democratic and Republican congressional staffers said the first step would require some sign-off from Senate Minority Leader Mitch McConnell (R-Ky.) and Senate Majority Leader Harry Reid (D-Nev.). Even without complete agreement, progress could be made if McConnell agrees not to obstruct a scaled-down deal such as the one proposed Friday afternoon by President Barack Obama. The president signaled Friday that he may ask Reid to push that proposal as a last resort.

The process could be helped along if Reid takes the suggestion made by McConnell and House Speaker John Boehner (R-Ohio) to start with one of the pieces of legislation already passed by the House.

Reid has steadfastly refused to do so, favoring a Senate-passed bill to preserve Bush-era tax cuts for income under $250,000. That bill, however, faces procedural objections from the House, which under the Constitution is supposed to originate tax measures.

If the Senate instead took up any of the bills the House has passed, the procedural problem would be solved. Reid could then essentially gut that legislation and replace it with Obama's proposal or whatever plan he and McConnell agreed should at least receive a vote without the threat of a filibuster. That bill could pass the Senate relatively quickly and be sent back to the House. Both Reid and McConnell suggested after their White House confab that they may be prepared to propose legislation Sunday -- less than two days before the pending cliff.

Indeed, at nearly 8 p.m. on Friday, Reid placed on the Senate calendar a House bill that would extend all the Bush-era tax cuts. It could serve as a vehicle to carry Obama's proposal or something other deal that Reid and McConnell work out over the next 24 hours to 36 hours.

This is where the equation becomes more complicated. Either Boehner would have to convince Republican House members to hold their noses and let the measure go to a vote -- even though they rebelled last week over his Plan B fallback, which included a much smaller tax hike than Senate Democrats or the president would back -- or he would have to ignore his caucus and simply schedule a vote.

A GOP House leadership aide suggested that Boehner might indeed go with the latter option. "The Speaker told the President that if the Senate amends the House-passed legislation and sends back a plan, the House will consider it -- either by accepting or amending," the aide emailed. "The group agreed that the next step should be the Senate taking bipartisan action."

Senate leaders sounded cautiously optimistic that this path, in some form or another, would be taken.

"We had a good meeting down at the White House," McConnell said in a statement. "We are engaged in discussions -- the Majority Leader and myself and the White House -- in the hopes that we can come forward as early as Sunday and have a recommendation that I can make to my Conference and the Majority Leader can make to his Conference. And so we'll be working hard to try to see if we can get there in the next 24 hours. So I'm hopeful and optimistic."

"Whatever we come up with is going to be imperfect," Reid said. "Some people aren't going to like it; some people will like it less. But that's where we are. We have an obligation to do the best we can."

In a later statement, the Senate majority leader said he was preparing Obama's backup measure. "At President Obama's request, I am readying a bill for a vote by Monday that will prevent a tax hike on middle-class families making up to $250,000, and that will include the additional, critical provisions outlined by President Obama," Reid said. "In the next twenty-four hours, I look forward to hearing any good-faith proposals Senator McConnell has for altering this bill."

If the last-ditch efforts all fail, partial success in the Senate would at least signal to observers that a deal is not that far off, albeit on the other side of the cliff. At that point, income taxes would have returned to the Clinton-era rates, and a proposal like the Senate's $250,000 plan would technically be a tax cut, instead of a hike.

The fiscal cliff's mandatory spending cuts -- the so-called sequester of $1 trillion over a decade -- would still need to be dealt with, but their impact would be less immediate and less likely to quickly slow the economy.

UPDATE: This article has been updated to note that Reid on Friday night placed on the Senate calendar a House bill to extend all the Bush-era tax cuts.

Michael McAuliff covers Congress and politics for The Huffington Post. Talk to him on Facebook.

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