A crucial piece of President Biden’s domestic agenda hung in the balance Thursday, as Democratic leaders moved toward a planned House vote on a $1 trillion infrastructure bill that many progressive Democrats have threatened to oppose.

Congress did appear poised to avert a government shutdown, with the Senate and then the House expected to easily approve a measure extending government funding through Dec. 3. Without passage of that bill, funding would expire and a partial government shutdown would begin at 12:01 a.m. EDT Friday.

But the infrastructure measure’s fate was far less clear, as Democratic leaders tried to unify the party’s increasingly combative progressive and moderate wings around changes to a separate $3.5 trillion healthcare, education and climate package.

Progressives want to see the larger measure advance as a condition of supporting the infrastructure bill in the narrowly divided House. Though 19 Senate Republicans backed the infrastructure bill and at least some House Republicans are also expected to do so, it isn’t clear whether there will be enough GOP support to offset opposition from liberal Democrats.

As of Wednesday night, House Speaker Nancy Pelosi (D., Calif.) was sticking to her plan to bring the infrastructure bill up for a vote on Thursday. She said she was taking it “one hour at a time,” though earlier in the day she opened the door to further delay, if talks didn’t progress.

Rep. Josh Gottheimer (D., N.J.), a centrist who had pushed for a vote on the infrastructure bill this week, said late Wednesday he still expected it to occur.

“At the end of the day, I still don’t believe a faction of the party would kill a key part of the president’s agenda,” he said.

Many of Mr. Biden’s most sought-after domestic-policy proposals are included in either the infrastructure or reconciliation measure. The president met Wednesday evening with Mrs. Pelosi and Senate Majority Leader Chuck Schumer (D., N.Y.) at the White House. Mr. Biden also met with moderate Democrats in recent days, seeking to lock down their support for the social-policy and climate bill, an effort that could mollify progressive fears that moderates would block it.

That endeavor has so far fallen short: Key centrist Sen. Joe Manchin (D., W.Va.) said on Wednesday that he didn’t think he could reach an agreement with the White House soon. Democrats need all 50 senators in their caucus to remain united to pass the $3.5 trillion package through a process called reconciliation. It requires just a simple majority rather than the 60 usually needed to advance in the chamber.

In a lengthy statement, Mr. Manchin repeated his concerns about additional spending fueling inflation and called for the bill’s measures to be means-tested. He didn’t outline a possible compromise with other Democrats.

“While I am hopeful that common ground can be found that would result in another historic investment in our nation, I cannot—and will not—support trillions in spending or an all-or-nothing approach that ignores the brutal fiscal reality our nation faces,” Mr. Manchin said.

Mr. Manchin’s statement sparked outrage among liberal House Democrats, who said that it had expanded the ranks of lawmakers willing to oppose the infrastructure bill, if a vote were held Thursday. Progressives see that a threat to oppose the infrastructure bill in the House could pressure moderate Democrats, particularly Mr. Manchin and Sen. Kyrsten Sinema of Arizona, to agree to the contours of the education, healthcare and climate package.

“This is why we’re not voting for that bipartisan bill until we get agreement on the reconciliation bill, and it’s clear we’ve got a ways to go,” said Congressional Progressive Caucus Chairwoman Rep. Pramila Jayapal (D., Wash.).

The possibility that the infrastructure bill might not pass Thursday frustrated some centrists who had secured the initial promise from Mrs. Pelosi of a vote this week.

“If the vote were to fail tomorrow or be delayed, there would be a significant breach in trust that would slow the momentum in moving forward in delivering the Biden agenda,” said Rep. Stephanie Murphy (D., Fla.), a co-chair of the centrist Blue Dog Coalition.

Some lawmakers said that delaying the vote would simply reflect that Democrats needed more time to wrap up complex negotiations.

“To me, it’s a bump in the road,” said Rep. Tom O’Halleran, a centrist Democrat from Arizona. “We’re going to get it done within a reasonable time,” he said of the infrastructure bill.

Failing to pass the infrastructure bill, which reauthorizes the federal transportation programs, before Friday at 12:01 a.m. EDT would put several thousand federal employees on furlough, according to the Transportation Department. Lawmakers are discussing a short-term extension of surface transportation authorization in the event that the infrastructure bill doesn’t pass Thursday, according to Sen. Shelley Moore Capito (R., W.Va.).

In addition to the fracas over Mr. Biden’s policy agenda among Democrats, Republicans and Democrats are also locked in a stalemate over raising the country’s borrowing limit. Treasury Secretary Janet Yellen told lawmakers on Tuesday that the country would be unable to pay its bills starting Oct. 18 unless Congress acts.

Republicans have blocked Democratic attempts to suspend the debt limit in the Senate, protesting the scope of Democrats’ spending ambitions and arguing that Democrats carry the responsibility for authorizing more borrowing.

Democrats have accused Republicans of creating the risk of a potentially catastrophic default on the debt, offering to pass the measure along party lines, if Republicans first allow the process to move forward. Democrats do have the power to raise the debt limit through reconciliation. So far they have resisted going that route, calling reconciliation time-consuming and unnecessary.

Democrats had originally paired the debt-limit measure and the government-funding patch together, trying to raise pressure on Republicans to support the must-pass measures. Republicans still blocked the bill, which also includes $28.6 billion in emergency disaster aid and $6.3 billion to help resettle Afghan evacuees.

Removing the debt-limit measure from the government-funding patch helped secure Republican support for the latter bill, which was expected to move relatively quickly through both chambers Thursday.

The two parties will continue to battle over the borrowing limit. The House approved a suspension of the debt limit on Wednesday in a 219-212 vote, with two House Democrats opposing it and one Republican voting in favor.