As bipartisan debt-reduction talks continue — and by all accounts, accelerate — participants are apparently down to the most challenging part of the process. Vice President Biden told reporters yesterday, “Now we’re getting down to the real hard stuff: I’ll trade you my bicycle for your golf clubs.”

The key players on both sides will reportedly start working “around the clock” next week to resolve the differences, with the goal of having a tentative agreement before the 4th of July.

We don’t yet know what an eventual deal might look like, but conservative Washington Post columnist Michael Gerson let readers know what he’s heard.

The most consequential recent political debate did not take place in New Hampshire. It is being held at Blair House, where Vice President Biden is convening bipartisan negotiations for an increase in the debt limit. Biden, by most accounts, is conducting those talks fairly. House Majority Leader Eric Cantor describes them as “constructive.” Important, if true.

For the first time, Senate Republicans describe to me the outlines of a possible deal: a package of immediate and specific budget cuts; budget caps reaching out five years to reassure conservatives that tough budget decisions will be made in the future; Medicare reforms short of the House approach; no tax increases — a Republican red line — but perhaps additional revenue from the elimination of tax expenditures.

Senate Minority Leader Mitch McConnell seems particularly intent on reaching a Medicare agreement different from Rep. Paul Ryan’s proposal. An incremental approach would allow Republican senators to say they confronted the problem, while allowing them to distance themselves from the unpopular House plan.

As is always the case, the devil will be in the details. Some additional revenue now seems like an inevitability, though it won’t come from literal tax increases, but rather an end to industry breaks and loopholes (see “ethanol, subsidies for”). The talk of caps is enough to make me break out in a cold sweat, though again, the specifics of the policy matter.

But what I found especially interesting about Gerson’s take was the notion that McConnell wants a Medicare deal that won’t resemble Paul Ryan’s privatization plan. That’s not surprising — the GOP had to know Dems would never, ever go along with this — but what is surprising is McConnell’s sense that his members will somehow get political cover on the issue. As Gerson put it, once Republicans embrace a different Medicare policy, they’ll be able to say they “confronted the problem,” while distancing themselves from the House plan.

This isn’t much of a strategy. Last month, the House budget plan came to the Senate floor and 40 Republicans voted for it. When Dems run campaign ads against all 40, telling voters these GOP senators voted to “end Medicare,” and replace it with a privatized voucher scheme, those ads will be accurate.

Even if this bipartisan deal, if it comes together, takes a different route, this vote won’t disappear.

If McConnell didn’t want to leave his caucus on the hook for supporting Medicare privatization, he should have thought of that before May 25. Deal or no deal, this vote will remain an albatross around the GOP’s neck.

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Follow Steve on Twitter @stevebenen. Steve Benen is a producer at MSNBC's The Rachel Maddow Show. He was the principal contributor to the Washington Monthly's Political Animal blog from August 2008 until January 2012.