The tax law, the GOP's signature achievement of this Congress, has failed to catch on with voters the way Republicans hoped. About 37 percent of registered voters approve of the platform, versus roughly 42 percent who disapprove, according to a RealClearPolitics average of polls . Consequently, Republican candidates have not used it heavily in ads across the country, despite the insistence from House Speaker Paul Ryan and other GOP leaders that the measure is working.

Republicans have tried to put tax reform back to the forefront in recent days, about two months ahead of midterm elections when they will try to stop Democrats from taking a House majority. Later this month, the House GOP aims to pass a bill that would make individual tax cuts approved last year permanent and would ease rules around retirement accounts, among other proposals.

"They are running on it and it's a huge mistake because no one wants to go back to the bad old days when the economy was flat, paychecks were flat," Rep. Kevin Brady , a Texas Republican, said in an interview that aired Friday morning on CNBC. He chairs the House Ways and Means Committee, the chamber's tax writing panel.

Voters haven't embraced the tax overhaul President Donald Trump signed late last year, but one of the main architects of the bill still thinks Democrats made an error in pledging to revisit the law.

Democrats have contended the plan, which slashed the tax rate for corporations from 35 percent to 21 percent, benefits businesses and the wealthy more than workers. They also point to the $1 trillion or more the plan is expected to add to budget deficits over a decade, even after economic growth is taken into account.

House Minority Leader Nancy Pelosi has said she wants to reassess the GOP-crafted bill on a bipartisan basis and potentially repeal parts of it. Multiple Democratic House candidates running in Pennsylvania and New Jersey races important to the battle for the chamber have also told CNBC they want to revisit the tax law, if they get elected.

With the new phase of tax reform, Republicans partly aim to put Democrats in a box. If Democrats oppose the measure in a vote, the GOP may argue Democrats did not really care about cutting rates for the middle class. The individual tax cuts passed under last year's plan are set to expire after 2025.

Passing the plan would not help the GOP with deficit concerns. The cost of making individual cuts permanent is an estimated $627 billion over a decade, according to the bipartisan Joint Committee on Taxation.

It could also create more problems for some incumbent House Republicans, as it would make the $10,000 cap on state and local tax deductions permanent. The provision pushed a dozen House GOP lawmakers in high-tax blue states to vote against their party's bill last year.

One of those representatives, Leonard Lance of New Jersey, told CNBC's "Power Lunch" that he supports "full deductibility" of state and local taxes. On Friday, he also contended that passing the second set of tax cuts "would be an exercise in futility because it will never pass the Senate." That would require at least nine Democratic votes in the Senate.

In a statement, Democratic Congressional Campaign Committee spokesman Tyler Law said, "The reality is that Republicans said in their own words that selling the tax bill is necessary to holding the House, but they have failed miserably as the bill continues to be less and less popular."

A wide range of Democratic candidates — including Rep. Conor Lamb, who won a special election in a red Pennsylvania district earlier this year — have also used the tax cuts to stoke concerns that the GOP could trim back Social Security and Medicare to make up for budget deficits.