Tax code books are stacked on the dais as Chairman Kevin Brady (R-Tex.) presides over the first day in the House Ways and Means Committee’s markup of the GOP’s tax plan on Nov. 6. (Melina Mara/The Washington Post)

Congressional Republicans face critical decisions this week as they move within striking distance on a major legislative package to cut taxes, an achievement party leaders say is crucial to stabilizing the GOP's recent political tailspin ahead of next year's elections.

The House plans to vote on a GOP tax bill by week's end that would slash taxes for companies and overhaul the tax code for virtually every American family and individual. And the Senate Finance Committee expects to vote on its version of the package within the next few days.

Stark differences between the House and Senate tax bills remain unresolved, but there is enough overlap and — so far — muted intraparty resistance, making the White House increasingly optimistic that an agreement can come by Christmas, as President Trump has repeatedly promised.

Enactment of the tax-cut package would mark Trump's first major legislative accomplishment at a time when his job approval rating has dropped to a record low for this point in a presidential tenure.

Still, potential quagmires remain. House and Senate Republicans risk colliding over whether Americans should be able to deduct local property taxes from their federal taxable income. The House GOP bill would allow Americans to deduct up to $10,000 of those taxes from income as a way to placate complaints from conservatives in high-tax states such as New York, New Jersey and California.

In an interview on "Fox News Sunday," Rep. Kevin Brady (R-Tex.), chairman of the House Ways and Means Committee, stood firm on that provision, saying it is important to "make sure people keep more of what they earn, even in these high-tax states."

Americans are permitted to use local property taxes as well as state income taxes to offset parts of their federally taxed income. Senate Republicans in their proposal have refused to allow for the deduction of any of those taxes.

Meanwhile, House and Senate bills differ importantly on how a huge corporate tax cut would go into effect. In the House GOP bill, the tax rate for corporations would fall from 35 percent to 20 percent in 2018. In the Senate bill, the tax cut wouldn't take effect until 2019, delaying some benefits for corporations but shaving more than $100 billion off the cost of the change.

Congressional Republicans have not indicated how they plan to address this discrepancy, but finding agreement on the timing of the cuts and how to treat property tax income would resolve some of the biggest outstanding issues.

Ten months after Republicans took control of the White House and Congress, the imperative could hardly be greater for a party that repeatedly fell short of its promise to repeal and replace the Affordable Care Act.

The results of Tuesday's election — in which Democrats swept governor's races in Virginia and New Jersey, as well as other contests around the country — have heightened the need for the GOP to demonstrate that it can govern. So, too, has the controversy that has enveloped Roy Moore, the GOP Senate candidate in Alabama, who has been accused by women of pursuing them when they were teenagers and he was in his 30s.

[Woman says Roy Moore initiated sexual encounter when she was 14, he was 32]

Failure to deliver on tax cuts, a growing number of Republicans say, could have consequences in races across the country next year, particularly if GOP donors close their wallets and the enthusiasm of grass-roots volunteers is sapped by the inaction.

"They have to pass something they can call tax reform or get under their desks and wait for the shelling," said John McKager "Mac" Stipanovich, a longtime Florida-based GOP consultant. "The perceived level of dysfunction in a Republican-led Congress and the perceived level of incompetence in a Republican-led White House would have enormous consequences in the 2018 election."

Asked Sunday about the impact of Tuesday's Democratic victories, Marc Short, Trump's legislative affairs director, suggested a simple antidote.

"The president and we believe that what we need to do is deliver on the tax relief that we promised," he said during an appearance on NBC's "Meet the Press."

More broadly, Republicans are battling criticism that their legislation would disproportionately benefit corporations and the wealthy, giving less relief to the middle class and in some cases pushing up taxes for middle-class families.

Trump and GOP leaders have promised for months that all Americans will receive a tax cut under their plan, but in recent days, GOP leaders have conceded that these were false statements and that some people's taxes will actually rise.

Analysts believe that most people will see their taxes fall, but millions of people will still pay more under the plans as they are currently constructed.

Democrats pounced on that notion Sunday.

"The reality is many middle-class families are going to end up paying more," Sen. Chris Van Hollen (Md.), chairman of the Democratic Senatorial Campaign Committee, told Fox News. "Folks in suburbs are going to get clobbered."

Steven Mnuchin, Trump's treasury secretary, stressed that most middle-income Americans will get a tax break under both GOP plans pending in Congress. But he stopped short of saying everyone would get a break.

"For most people — and, again, it may not be 100 percent, but by far the majority — both the House and Senate version provide middle-income tax relief," Mnuchin said on CNN's "State of the Union."

While Republican strategists widely agree on the need to act on taxes, some fret that the party's message has already become too muddled. Marc Rotterman, a veteran GOP consultant in North Carolina, said he would have preferred to see a less complicated bill that more closely matched Trump's campaign rhetoric.

"This is not what a lot of us thought it would be," Rotterman said. "I think they're listening too much to K Street."

But the imperative to score a win remains powerful.

Rep. Chris Collins (R-N.Y.) said he's being pressed by everyone from GOP donors to rank-and-file voters.

"Republicans of all ilk, my supporters, everyone is saying we need to get tax reform done. We didn't get health care done," Collins said. "The point is we've got to deliver this to keep our base enthused to turn out in the midterms."

A number of Republicans are pushing for changes to the House and Senate tax bills, primarily eyeing ways to ensure that the legislation helps working families.

The Tax Policy Center found last week that most families would see tax cuts under the House Republican plan, but it did say that 7 percent of families would see tax increases in 2018 and 25 percent would see tax increases in 2027.

This increase is based in part on a House provision that expires after five years that allows each adult to claim a $300 credit against their income. House Republicans decided to cap this credit after five years in part to extend the corporate tax cut they wanted to deliver.

Rep. Warren Davidson (R-Ohio), a conservative member of the House Freedom Caucus, wrote in a Twitter post Friday that "#TaxReform that raises your taxes is the wrong direction. Still reviewing scenarios from the House and Senate bills. . ."

On the other side, Republican Sens. Bob Corker (Tenn.), Jeff Flake (Ariz.) and James Lankford (Okla.) have all raised concerns about additional debt from a tax-cut measure. The House bill would add roughly $1.4 trillion to the debt over 10 years, and the Senate bill would add $1.5 trillion, according to estimates from the Joint Committee on Taxation.

Meanwhile, Senate aides are still sorting through an extremely problematic component of their tax bill.

Senate Republicans aim to pass their tax bill along party lines, without any support from Democrats, and that requires them to use a process called reconciliation.

But to pass a tax-cut bill through reconciliation, Senate rules say, the legislation cannot add to the debt after 10 years. The Senate GOP tax bill would add to the debt after 10 years, and officials have to come up with a way to address this, or the bill will essentially be disqualified from becoming law.

That leaves them with some unpopular choices. They can decide to have major parts of the tax-cut package expire after 10 or fewer years, or they can add some tax increases that would not kick in until the late 2020s, assuming Congress will eventually vote to reverse the hikes.

"If they thought they had the solution, it would have been in the [bill] already," said Douglas Holtz-Eakin, a conservative economist and former director of the Congressional Budget Office.

Dino Grandoni and Mike DeBonis contributed to this report.