WASHINGTON — Amid the swirl of recriminations and second-guessing that followed the announcement by Senator Joe Manchin III that he would not support President Biden’s signature domestic policy plan, one point of contention stood out: their differences over an expansion of the child tax credit.

The legislation would extend a program under which most families have been receiving monthly payments of up to $300 per child. The program had long been seen by most Democrats as a cornerstone of their social welfare plan, hoisting millions of children out of poverty while also giving new support to the middle class. And they had hoped to make it a central plank of their midterm campaigns, evidence that Democrats could deliver real help to struggling families.

But in opposing the sprawling climate, social spending and tax bill as written, Mr. Manchin, a West Virginia Democrat, has been clear about his deep qualms with the cost and structure of the current credit, vacillating between rejecting an extension outright or demanding new limits such as work requirements for parents.

“We have one chance at this, OK?” Mr. Manchin said on Monday, in an interview with a West Virginia broadcaster, when asked about his rationale for opposing a package that could benefit his state. Mentioning the child tax credit, he added that he preferred to “make sure that people get it that need to get it.”

Without passage of any legislation before the end of the year, families will have received their last payment under the current plan on Dec. 15, losing a stream of money that has helped parents cobble together healthier meals, facilitate car repairs and cover utility bills. It could be particularly devastating for families in Mr. Manchin’s home state of West Virginia, where an estimated 50,000 children could slip deeper into poverty.

“Having this be a cash benefit that can go to food if it’s needed, or clothing if it’s needed, or build savings — that flexibility is really important,” said Kelly Allen, the executive director of the West Virginia Center on Budget & Policy. “Families know what they need.”

Resolving the differences between the administration and Mr. Manchin over the proposal, all while retaining the support of nearly every congressional Democrat — Republicans are opposed — will be crucial for party leaders as they seek to revive the package next month, when Congress is scheduled to return from its winter recess.

Mr. Biden, pressed on Tuesday about whether a deal could be salvaged, sought to downplay his administration’s frustrations with Mr. Manchin and insisted that he believed there remained a chance for the package, known as the Build Back Better Act, to become law. He named the child tax credit as a crucial element that would help parents across the country.

“Senator Manchin and I am going to get something done,” he concluded.

Senate Democrats were set to convene a rare evening caucus call on Tuesday to discuss their path forward, with both the domestic policy package and voting rights legislation in limbo without Mr. Manchin’s support.

Congress expanded the scope of the child tax credit through the end of the year as part of the $1.9 trillion pandemic relief law, which cleared Congress with just Democratic votes, including Mr. Manchin’s. Democrats hope to extend it and ultimately make it permanent.

The credit, first established in the late 1990s, was increased to cover more families and restructured to allow direct monthly payments this year of up to $250 a child, or $300 for a child under 6. It also became fully refundable, meaning families did not need to provide proof of income or owe federal income tax to receive the payment.

Of the country’s 74 million children, nearly nine in 10 qualified, and the expanded eligibility for families at the low end of the income scale significantly reduced child poverty.

In an October Census Bureau report that tracked how about 300,000 recipients spent their first three payments this summer, researchers found that about half of families spent part of the money on food. Other families said they used the money to help pay their rent, mortgage, utilities, child care and school expenses.

Under the version of the domestic policy bill passed by the House, the program would be extended by another year. Without an extension of the expanded program, the size of the credit will fall substantially, especially for low-income families. And it will revert to being paid in a once-a-year lump sum connected to filing a tax return rather than half of it distributed monthly.

As Democrats hailed the payments hitting their constituents’ bank accounts this year, Mr. Manchin balked at the cost of the one-year extension, arguing that it was misleading given that there were ambitions to make it permanent, a step that would cost about an additional $1.5 trillion over a decade, according to the Congressional Budget Office.

At one point, Mr. Manchin also privately voiced concerns to his colleagues that the payments could help fuel opioid use, comments that were first reported by The Wall Street Journal and confirmed by a person familiar with the discussion. Other Democrats and supporters of the proposal have rejected that concern, and argued that steps suggested by Mr. Manchin, such as imposing work requirements and additional limits on who could receive the money, would make it harder for the neediest families to receive the full benefit.

On Tuesday, Mr. Manchin went to the White House and put forward his own outline for the domestic policy plan, which included money for universal prekindergarten, child care and some environmental provisions, but did not include the one-year extension of the child tax credit, according to people familiar with the offer.

In the local interview on Monday, Mr. Manchin reiterated his concern with the scope of the plan, adding that his preference was to “make sure that people that are working. If it’s called a tax credit, you got to have a W-2.”

Supporters of the plan in his home state said his position was at odds with the needs of many of his constituents.

“I don’t think there’s a way to really explain what it means to feel like you’re finally getting your head above water, and then have that yanked out from underneath you,” said Amy Jo Hutchison, a West Virginia organizer who has been working with advocacy groups to lobby Mr. Manchin to support the package. “People were really starting to breathe a little easier.”

It remains unclear whether Mr. Biden and other Democrats in Congress would give in to Mr. Manchin and curtail or remove the tax credit, after months of vowing to make it permanent. Without Mr. Manchin’s support, Democrats have little chance of passing the broader legislation or pushing the child tax credit extension as a stand-alone bill, since there is little indication Republicans would support it as written.

Pressed on whether her caucus would accept scaling back the credit or imposing work requirements, Representative Pramila Jayapal of Washington, the chairwoman of the Congressional Progressive Caucus, declined to answer, telling reporters on Monday, “I am not going to negotiate with Joe Manchin on something that I don’t even know he wants to do.”

Mr. Manchin’s refusal to take up the plan is drawing pushback beyond progressives, including from the country’s largest coal miner’s union, which called on the senator to reconsider.

“We urge Senator Manchin to revisit his opposition to this legislation and work with his colleagues to pass something that will help keep coal miners working, and have a meaningful impact on our members, their families and their communities,” Cecil E. Roberts, the president of the United Mine Workers of America, said in a statement.

The version passed by the House included tax incentives that encourage manufacturers to build facilities in coal fields. It also extended an excise tax that is the only source of revenue for a trust fund that pays benefits to coal miners with black lung disease as well as their survivors when the employer is unable to provide the benefits.

“Senator Manchin has always been a strong advocate for the U.M.W.A. and led legislation to address the black lung excise tax expiration,” said Sam Runyon, a spokeswoman for Mr. Manchin. “He will of course continue to work to shore up the black lung excise tax in the new year to address the needs of our brave miners.”

Lisa Friedman contributed reporting.