WASHINGTON — President Biden’s pledge to fully pay for his $1.85 trillion social policy and climate spending package depends in large part on having a beefed up Internal Revenue Service crack down on tax evaders, which the White House says will raise hundreds of billions of dollars in revenue.

But an upcoming assessment by the nonpartisan Congressional Budget Office is likely to undercut the administration’s position by casting doubt on the amount of money that a more muscular I.R.S. would actually bring in.

As Democrats prepare for a final push to pass the spending legislation this week, the White House is bracing lawmakers for a disappointing tally from the budget office, which is likely to find that the cost of the overall package will not be fully paid for with new tax revenue over the coming decade. That includes the administration’s plans to boost I.R.S. funding by $80 billion, which Treasury says could bring in $400 billion over the next decade as a bolstered enforcement staff ramps up audits on corporations and the rich. The budget office is expected to assume a much lower haul, according to White House and Treasury officials — possibly less than $200 billion over 10 years.

The C.B.O. tends to believe that the tax collection prowess of more enforcement agents will wane over time, while the White House assumes that taxpayers will become more compliant with the I.R.S. when they see tax dodgers facing consequences.

Such estimates are crucial to Mr. Biden’s ability to get the next leg of his agenda through Congress. Lawmakers have to rely on the budget office’s so-called score, which estimates whether the spending will add to the federal budget deficit over the next 10 years.

A disappointing assessment that shows the bill adding to the deficit could pose another big challenge for Mr. Biden’s domestic policy legislation, which is already facing steep hurdles in the House and Senate. A group of moderate Democrats in the House have said that they want to see an assessment from the budget office before moving forward with the legislation. And some lawmakers have expressed concerns about whether the bill is fiscally responsible, with Senator Joe Manchin III of West Virginia, a key swing vote, expressing concern that the package could add to the national debt and stoke further inflation.

Because Democrats are using a budget procedure called reconciliation to pass the bill with a simple majority, they cannot afford to lose a single vote in the Senate and no more than three votes in the House.

The administration’s ability to raise taxes to pay for the spending has already run into resistance. Mr. Manchin and other moderate Democrats have opposed efforts to sharply raise taxes on corporations and the wealthiest Americans. That has left the Biden administration increasingly reliant on capturing uncollected tax revenue from the $7 trillion “tax gap” to pay for a sweeping expansion of child care, health and climate initiatives.

The proposal to give the I.R.S. an additional $80 billion over a decade has drawn fierce resistance from Republicans, right-leaning advocacy groups and banks, who have warned that an empowered tax collection agency will be weaponized against conservatives and target ordinary taxpayers.

The Biden administration has insisted that audit rates for people earning less than $400,000 per year would not rise, but that a large expansion of the nation’s social safety net could be funded just by collecting tax revenue that is already owed to the government.

The big question is: How much money is there for the taking?

A preliminary assessment by the budget office earlier this year suggested the administration was being overly optimistic and that those who had avoided paying taxes in the past would adjust their activities to continue evading the I.R.S.

“C.B.O. expects taxpayers to adapt to the I.R.S.’s enforcement activities and adopt new ways of evading detection, so an enforcement activity may have a lower return in later years,” the budget office said in September.

Bracing for the shortfall, senior Biden administration officials are urging lawmakers to disregard the budget office assessment of the enforcement proposal. They argue that the budget office is being overly conservative in its calculations, failing to properly credit the return on investment of additional I.R.S. resources and overlooking the “deterrent effects” that a more aggressive tax collection agency would have on tax cheats.

“In this one case, I think we’ve made a very strong empirical case for C.B.O. not having an accurate score,” Ben Harris, Treasury’s assistant secretary for economic policy, said in an interview. “The question is would they rather go with C.B.O. knowing C.B.O. is wrong or would they want to target the best information they could possibly have.”

Mr. Harris described the discrepancy as a methodological shortcoming. He said that it was “patently absurd” that bolstering the enforcement capacity of the I.R.S., which has been depleted for years, would not compel taxpayers to be more compliant. The C.B.O. also predicts that the “return on investment” of giving the I.R.S. more money will decline over time, while Treasury disagrees.

The C.B.O. has been releasing its assessments of the House Democrats’ legislation in parts and has been racing to get an overall number to lawmakers ahead of a possible vote this month. Most of the estimates are expected to be in line with White House projections, but the I.R.S. measure is likely to be an outlier.

The I.R.S. has for years been a favorite target of Republicans, who have accused the agency of political bias and worked to starve it of funding. From 2010 to 2020, funding for the I.R.S. declined by about a fifth and its enforcement ranks fell by 30 percent, making it difficult to pursue audits and legal fights against well-financed tax evaders.

In recent weeks, Republicans in congress have expressed growing alarm about the prospect of an empowered I.R.S.

“The I.R.S. will double in size,” Representative Mike Kelly, Republican of Pennsylvania, said last month. “It will be more involved in the day-to-day lives of every American. And the result will be an invasion of privacy and the heavy hand of the government squeezing out smaller, more local businesses.”

The Biden administration believes that doubling the enforcement staff at the I.R.S. will go a long way toward combating tax dodgers.

Charles P. Rettig, the I.R.S. commissioner who was picked for the job by former President Donald J. Trump, said last week the agency was long overdue for a financial infusion. He said the agency has fewer auditors than at any time since World War II.

“If given the resources we need, we will be able to make a sizable dent in noncompliance over several years,” Mr. Rettig wrote in a Washington Post opinion article. “A properly funded and trained workforce will also have a significant deterrent effect on cheating.”

A separate proposal that would also have required banks to report more information about the finances of their customers to the I.R.S. has so far been left out of the legislation amid backlash over privacy concerns. The Biden administration is still pushing for a more narrow version of that proposal to be included in a final bill.

Douglas Elmendorf, who directed the C.B.O. from 2009 to 2015, said that estimating the returns on additional I.R.S. enforcement was challenging because large funding infusions to the agency had little precedent and it was difficult to quantify the “indirect effects” of more auditors. He said lawmakers should take that into account when setting policy.

“I think Congress should always look beyond the budget estimate when deciding what to do about legislation,” Mr. Elmendorf said.

With slim majorities in the House and Senate, Democrats could need to find other ways to pay for their plans if they are not ready to rely on the I.R.S.

John Koskinen, the former I.R.S. commissioner in the Obama and Trump administrations, said that it was unfortunate that the proposals to fund the agency became so politicized. He suggested that it was not so far-fetched that an agency that already collects more than $3 trillion a year could capture another $40 billion annually if it was properly staffed and modernized.

“When you under-fund the I.R.S., it’s just a tax cut for tax cheats,” Mr. Koskinen said.