Speaker of the House Paul Ryan (R-WI) speaks to the media after a House Republican conference on Capitol Hill in Washington.

The plan comes out two months ahead of November's midterm elections, where Republicans are trying to prevent Democrats from winning a House majority largely by arguing their policies have boosted economic growth. The GOP tax plan passed last year, which made steep, permanent cuts to the corporate tax rate, has not caught on with the public the way the party had hoped.

The House has a busy month ahead of it, as Congress tries to pass a spending bill by the end of September to avert a government shutdown.

Earlier in the day, House Speaker Paul Ryan said he hopes to hold a floor vote on the plan later this month. House Republicans were meeting Thursday to get briefed on the proposal.

The second stage of tax reform from House Republicans will aim to make individual cuts passed last year permanent and ease rules around retirement savings, according to details shared with CNBC on Thursday.

Under the plan passed last year, tax changes on the individual side including rate reductions and a doubled child tax credit were set to expire in 2025. The new proposal would lock in those provisions, as well as one of the most controversial parts of last year's bill. It would make permanent the $10,000 cap on state and local tax deductions, which prompted a dozen House Republicans in high-tax blue states to oppose the plan.

It would include provisions designed to make entrepreneurship easier, including one that would allow businesses to write off more of their start-up costs. It would also aim to encourage savings by reducing barriers to employers providing retirement plans.

The House GOP argues its new tax proposals will help boost economic growth and aid small businesses. House Ways and Means Committee Chairman Rep. Kevin Brady, R-Texas, told the GOP conference Thursday that "the best is yet to come." The party aims to make the tax code better every year it holds control in Washington, he said to colleagues.

It also creates another potential problem for the GOP: budget deficits. The cost of making individual cuts permanent is an estimated $627 billion over a decade, according to the bipartisan Joint Committee on Taxation.

Since the GOP tax plan passed last year, Democrats have repeatedly criticized the projected $1 trillion or more it is expected to add to deficits over a decade. On Thursday, Brady told House Republicans that he is convinced deficits would be higher if they do not pass the second round of tax reform.

Speculation had grown that Republicans could try to index capital gains taxes to inflation as part of their second piece of tax legislation, but the GOP did not include it. Brady told colleagues that it would be "incredibly pro growth" and that the party would have to have a "family discussion" about whether to take care of it in a third of fourth round of reforms.

Correction: House Speaker Paul Ryan said Thursday that he hopes to hold a floor vote on the plan later this month. An earlier version misstated the day.