WASHINGTON — In a pivotal week, in a make-or-break stretch for President Biden’s domestic agenda, congressional Democrats are trying to assemble a puzzle of four jagged pieces that may or may not fit together.

Making them work as a whole is critical for the party’s agenda and political prospects, and how quickly they can assemble the puzzle will determine whether the government suffers another costly and embarrassing shutdown — or, worse yet, a first-ever default on its debt that could precipitate a global economic crisis.

Here are all the moving parts.

Piece 1: Government funding.

At a second past midnight on Friday morning, the parts of the government that operate under the discretion of Congress’s annual spending process will run out of money if a stopgap spending bill does not pass. Oct. 1 is the beginning of the fiscal year, and with larger issues dominating their attention, the Democratic House and Senate have not completed any of the annual appropriations bills to fund the Departments of Defense, Transportation, Health and Human Services, State and Homeland Security, to name a few.

This is not unusual. More often than not, the individual funding bills do not pass until winter. In the interim, Congress passes “continuing resolutions” to keep departments open at current spending levels, with perhaps a few tweaks for urgent priorities and emergencies like hurricane response and, this year, Afghan refugee resettlement.

By Thursday, Congress could easily pass such a resolution to avoid a lapse in funding that could furlough federal workers and force “essential” employees, like those at the Transportation Security Administration, to work for no money. But on Monday, such a stopgap measure was blocked by Republicans in the Senate because it was attached to …

Piece 2: The debt limit.

The federal government has for decades operated under a statutory ceiling on the amount it can borrow — in common parlance, the debt limit. The $28 trillion federal debt climbs inexorably, not only because the government spends so much more than it recoups in taxes, but also because parts of the government owe money to other parts, mainly most of the government owing money from Social Security after decades of borrowing.

In essence, raising the debt limit is akin to paying off your credit card bill at the end of the month, because a higher borrowing ceiling allows the Treasury to pay creditors, contractors and agencies money that was already extracted from them in Treasury bonds and notes or contracts. It is not for future obligations.

The last time the issue surfaced, in August 2019, Congress and President Donald J. Trump suspended the debt limit through July 31 of this year. On Aug. 2, the Treasury reset the debt limit to $28.4 trillion, and the government crashed through it days later. Ever since, the department has been shuffling money from account to account to make sure its bills are paid, but sometime in mid- to late October, such “extraordinary measures” will be exhausted, and the bills will go unpaid. This would be a shock to the international economy, since U.S. government debt is a global safe harbor for all kinds of cash and investments.

During Mr. Trump’s presidency, Republicans and Democrats did not fight over debt limit increases, in part because large spending increases for the coronavirus pandemic and other priorities were bipartisan — although the large tax cut of 2017 was not.

This year, Republican leaders have declared that because Democrats control the House, the Senate and the White House, they and they alone must raise the debt ceiling.

Republicans have made it clear that they intend to filibuster an ordinary bill to raise the debt ceiling, as they did on Monday. For Democrats to do so unilaterally, they would most likely have to use a budget process called reconciliation that shields fiscal measures from a filibuster.

Doing so is a complex and time-consuming affair. It all has to be done in the next two to three weeks, to beat the still unknown but rapidly approaching “X date” when the government defaults.

Piece 3: Infrastructure.

In August, with rare bipartisan swagger, the Senate passed a $1 trillion bill to build or fortify roads, bridges, tunnels, transit and rural broadband networks. The 69 “yes” votes included Senator Mitch McConnell of Kentucky, the Republican leader, and 18 others from his party.

Then it got more complicated.

Pressing for a quick vote on the bill, nine conservative-leaning Democrats in the House threatened to withhold their votes for the party’s $3.5 trillion budget blueprint until the Senate-passed infrastructure bill cleared their chamber.

The budget blueprint was needed to move Mr. Biden’s sprawling social policy and climate change agenda past a Republican filibuster in the Senate, through the reconciliation process. So in a signature maneuver, Speaker Nancy Pelosi cut a deal with the nine moderates: Vote for the budget resolution to get the social policy bill underway, and she would take up the infrastructure bill by Sept. 27, three days before a host of existing transportation and infrastructure programs are to exhaust their legal authorization.

Ms. Pelosi hoped that by then, the reconciliation package would also be ready for action. But that has not happened, and now liberals in the House are threatening to withhold their votes for the infrastructure measure.

Sept. 27 came and went on Monday without a vote or a deal among the factions, with the speaker securing agreement from her moderates to put off action until Thursday. The question is whether enough liberal Democrats in the House will vote for it as they wait for the final details of …

Piece 4: Social policy reconciliation.

Democrats’ exceedingly ambitious social policy bill, which Mr. Biden calls his “Build Back Better” plan, is packed with longstanding party priorities. The House has drafted a 2,465-page version that includes a huge array of programs to combat climate change, the extension of a generous child tax credit, universal prekindergarten, greatly expanded access to community college, increased resources for elder care and paid leave, and a Medicare expansion to cover vision, hearing and dental care — all paid for by trillions of dollars in tax increases on corporations and the wealthy.

Ms. Pelosi had hoped to put it to a vote this week, but she faced two problems: As of now, Democrats most likely do not have the votes, and Senate Democratic leaders have yet to produce a detailed bill that can draw the support of every member of their caucus.

Several conservative-leaning Democrats in both chambers, including Senators Joe Manchin III of West Virginia and Kyrsten Sinema of Arizona, have said they cannot support the plan as proposed. And because Republicans have made it clear they are unified in their opposition, Democrats cannot afford to lose even one vote from their party in the Senate.

The math is almost as daunting in the House, where Democrats can afford to lose as few as three votes.

Mr. Biden has been negotiating with the holdouts to determine what they could support. But for now, the lack of agreement on the sprawling plan is blocking its progress — and leaving the fate of the infrastructure measure uncertain as well.

On Monday afternoon, Ms. Pelosi signaled to Democrats that a vote on the reconciliation plan would have to be pushed off until the differences were ironed out.