As Senate Democrats begin to strategize on the across-the-board spending reductions set to hit next month and other fiscal issues on Tuesday at their retreat in Maryland, House Republicans are pointing fingers at the White House, claiming that the administration came up with the idea for the sequester cuts in the first place.

In truth, the cuts—now set at about $85 billion divided between defense and domestic discretionary programs—have a much more complicated and bipartisan history than Republicans may want to acknowledge. But regardless of the sequester’s origin, many lawmakers in both parties would prefer to replace, or at least mitigate, the pending cuts.

How exactly Congress might do that will be a primary focus as Democratic senators gather for two days behind closed doors in Annapolis. House Democrats will leave for a retreat in Leesburg, Va., on Wednesday for their own issues conference. While gun control and immigration issues are likely to be discussed at both conferences, sequestration will take center stage—and few expect any tidy solutions to emerge.

Replacing some of the sequester cuts with new taxes or revenues is something most Republicans oppose, and Democrats oppose moves to replace military cuts solely through making even more reductions in domestic programs. Democrats are expected to discuss strategies such as paying off the sequester incrementally, along with finding potential new revenue sources to do so, such as targeting tax breaks or subsidies enjoyed by oil and gas companies.

As the Democrats mull their options, House Republicans, led by Speaker John Boehner of Ohio, aim to highlight the fact that President Obama missed Monday’s deadline to send a budget to Congress—the fourth time he has done so in five years. The White House announced last month that the deadline would be missed, noting that the scuffle over a fiscal-cliff deal in early January contributed uncertainty over current fiscal-year appropriations and other baseline budget projections.

But that did not stop Republicans from pursuing a vote this week on a bill both blaming Obama for the nation’s debt and the sequester and requiring the president to submit a second, follow-up budget each year that estimates when and how a balanced budget will be achieved. The bill amounts to a messaging move, with little chance of passage in the Senate.

That refrain resurfaced on Monday, as Boehner took to the House floor and declared, “The president first proposed this sequester in 2011; he insisted that it be part of [that year’s] debt-limit agreement.”

Of course, what Boehner did not mention was that this is the very same “debt-limit” deal that was agreed to and ultimately supported by majorities in both chambers back in the summer of 2011 with passage of the Budget Control Act—a deal that Boehner even described at the time as giving him “98 percent of what he wanted.”

Still, congressional Republicans have long claimed that the first time a sequester plan was broached was during a visit to the Capitol on July 23, 2011, in which White House staffers met with Boehner aides and raised the idea.

Administration officials do not deny that a sequester plan was discussed at that meeting, but they argue that it was not the first time. An enforcement mechanism had been brought up in various negotiations, they say, dating from bipartisan talks with House and Senate members led by Vice President Joe Biden earlier in 2011.

Indeed, news accounts of earlier talks involving congressional leaders and Obama at the White House mention “triggers” that would force Congress to follow through with agreed-upon cuts.

But that will not necessarily stop Republicans from blaming the White House for the sequester, and that refrain could be heard on the House floor on Monday. As Boehner said, “One example of something the president’s budget could have addressed is his ‘sequester.’