House Speaker

Nancy Pelosi

(D., Calif.) and Senate Majority Leader

Chuck Schumer

(D., N.Y.) said in a joint statement Monday that suspending the debt limit through December 2022 reflects an appropriate length of time to cover the debt incurred by a roughly $900 billion bipartisan Covid-19 relief package passed last year under former President

Donald Trump.

“The Republicans are doing a dine-and-dash of historic proportions that hurts the American people and hurts our country,” Mr. Schumer said on the Senate floor Monday. “Both sides—both sides—have a responsibility to pay for the debt that we have already incurred. It has always been done in a bipartisan way.”

Raising the debt limit doesn’t authorize new spending, but rather allows the Treasury Department to issue new debt to cover spending that Congress already has authorized, including payments to bondholders, Social Security recipients and veterans.

Republicans have criticized Democrats for passing a $1.9 trillion Covid-19 relief package earlier this year and for their efforts to squeeze through a $3.5 trillion plan of social-welfare and climate provisions, both without GOP support. The statement linking the debt-limit suspension to last year’s Covid-19 relief is designed to put pressure on Republicans who might be nervous about rattling markets if the threat of a default grows larger, aides said.

Republicans have urged Democrats to attach the debt-ceiling measure to their $3.5 trillion package, which is moving via a special process called reconciliation that requires just a simple majority in the Senate. That would allow Democrats to adjust the debt ceiling without GOP votes. Democrats have said they don’t want to do that and noted they voted with Republicans to raise the debt limit under Mr. Trump’s administration.

In the Democratic-led House, the combined spending patch and debit-limit measure can pass even if no Republicans support it. But GOP opposition can derail the bill in the evenly divided Senate, where the measure will need 60 votes to clear procedural hurdles.

Senate Minority Leader Mitch McConnell (R., Ky.) reiterated Monday that Senate Republicans wouldn’t vote for legislation raising the debt limit.

“Their strategy for all this transformational borrowing, spending and tax hikes was deliberately designed to include no Republican input and to receive no Republican votes” Mr. McConnell said on the Senate floor. “Since Democrats decided to go it alone, they will not get Senate Republicans’ help with raising the debt limit.”

President Biden said in a tweet Monday that he supported Democratic leaders’ plan to combine the debt limit with a short-term spending bill that is also expected to include disaster aid and funding for Afghan refugees.

“This is a bipartisan responsibility, just as it was under my predecessor. Blocking it would be inexcusable,” Mr. Biden wrote on Twitter.

Republican

Sen. John Kennedy

of Louisiana said Monday he expected he would likely vote for the measure because it included disaster aid.

“I’m probably going to vote for it because my state needs the help,” he said, but predicted it wouldn’t pass the Senate. Mr. McConnell isn’t likely to change his strategy, Mr. Kennedy said. “He’s not bluffing on this.”

Lawmakers said Monday it wasn’t clear how Congress would resolve the fight over the debt limit, now officially intertwined with the government’s funding.

“I am worried because I know Sen. McConnell, when he makes these pronouncements, has a tendency to ignore the consequences,” Senate Majority Whip Dick Durbin (D., Ill.) said when asked if he was worried about the possibility of a default. “He’s done things like this, reckless things like this in the past, and endangered the traditions of the Senate and in this case, the economy of our country.”

The standoff has alarmed Wall Street analysts and business leaders, who in recent weeks have issued warnings about a rising risk of a technical default, in which the government might be unable to make all of its regular payments in full and on time. The threat of such a default could derail markets and hit U.S. economic growth.

The House is expected to vote later this week on the measure extending the government’s funding until early December and suspending the debt limit. The government’s current funding expires at 12:01 a.m. Oct. 1.

Write to Kristina Peterson at kristina.peterson@wsj.com and Siobhan Hughes at siobhan.hughes@wsj.com