WASHINGTON—Senate Majority Leader Chuck Schumer said in a letter to colleagues Monday that lawmakers must act to increase the federal government’s borrowing limit by the end of the week to avoid any repercussions, but Democrats didn’t lay out how they planned to pass a bill without Republican votes.

“Let me be clear about the task ahead of us: we must get a bill to the president’s desk dealing with the debt limit by the end of the week. Period,” Mr. Schumer (D., N.Y.) wrote.

Treasury...

WASHINGTON—Senate Majority Leader Chuck Schumer said in a letter to colleagues Monday that lawmakers must act to increase the federal government’s borrowing limit by the end of the week to avoid any repercussions, but Democrats didn’t lay out how they planned to pass a bill without Republican votes.

“Let me be clear about the task ahead of us: we must get a bill to the president’s desk dealing with the debt limit by the end of the week. Period,” Mr. Schumer (D., N.Y.) wrote.

Treasury Secretary Janet Yellen has told Congress that the limit must be raised by Oct. 18 or the government won’t be able to pay its bills. A default on the debt could have catastrophic financial consequences, and the pressure on both parties to resolve the issue will likely grow as Congress nears the deadline.

Mr. Schumer didn’t specify in his letter what procedural steps he planned to take to increase the debt ceiling, but said Democrats would discuss the issue at their closed-doors lunch Tuesday.

Republicans have blocked efforts to increase the debt limit on a bipartisan basis, saying Democrats should have to supply the votes on their own, given their decision to pass spending bills without GOP support. In a letter to President Biden Monday, Senate Minority Leader Mitch McConnell (R., Ky.) urged him to “engage directly with congressional Democrats” on raising the debt ceiling.

He emphasized that Democrats could raise the debt ceiling on their own by reopening the reconciliation process, which would allow them to move a bill through the 50-50 Senate with just a simple majority, rather than the 60 votes required of most legislation. “This is the path they will need to take,” Mr. McConnell said.

Mr. Biden, in remarks at the White House, said the U.S. “is a nation that pays its bills and always has.” He called on Republicans to “get out of the way” and let Democrats quickly raise the debt limit. Asked whether he could guarantee that the U.S. would be able to raise the debt ceiling before the deadline, he put the onus on Republicans: “No, I can’t. That’s up to Mitch McConnell.”

Democratic leaders have said trying to address the debt limit through a process tied to the budget known as reconciliation would be risky and time-consuming. However, it does remain an option for Democrats, who could decide to revise their current budget resolution either to increase the debt limit as a stand-alone bill or fold it into a broader social policy and climate package they are currently writing.

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While top Democrats have argued against using reconciliation to raise the debt limit, their staff has also consulted with the chamber’s nonpartisan parliamentarian to chart out the process for possibly doing so, according to a person familiar with the efforts.

The parliamentarian recently advised both Democrats and Republicans that raising the debt limit through reconciliation wouldn’t interfere with Democrats’ broader social-policy and climate bill, which the party is pursuing through reconciliation, the person said.

The new instructions for the fiscal year 2022 budget resolution, which has already been approved in both chambers, would have to move through the House as well as the evenly-divided Senate Budget Committee, where Republicans could block it by not showing up and denying the panel a quorum. But provided that Republicans show up and the bill gets a tie vote in the Budget Committee, a simple majority of the full Senate could vote to bring it to the floor, with Vice President Kamala Harris providing the tiebreaking vote.

The Senate would then have to hold a debate and a typically long marathon of amendment votes known as “vote-a-rama” on the new revised budget resolution. Lawmakers would then write the new legislation increasing the debt limit to a specific figure, and that bill would go through the same process.

—Gabriel T. Rubin contributed to this article.

Write to Kristina Peterson at kristina.peterson@wsj.com and Alex Leary at alex.leary@wsj.com