The GOP wants to approve a plan by the end of the year to dramatically lower individual and corporate tax rates, double the standard deduction and scrap certain taxes that largely affect wealthier Americans. The party released a framework of its proposal last month , but it lacked some details Republicans aimed to hash out before releasing a draft bill.

The plan's release will follow an expected House vote on the Senate-approved budget plan on Thursday. The Senate resolution unlocks a tool that would allow the tax bill to pass with only a simple majority of Republican votes in the Senate.

Speaker Paul Ryan, R-Wis., conducts a news conference in the Capitol after a meeting of the House Republican Conference on October 24, 2017.

President Donald Trump was attending the Senate GOP policy lunch on Tuesday afternoon following a fresh war of words with Sen. Bob Corker, R-Tenn. Corker, the Senate Foreign Relations Committee chairman, had already expressed concerns about deficits generated by tax cuts.

Once they release the bill, Republicans will still have to overcome obstacles. Among those, some Republicans have already expressed concerns about the potential budget deficit generated by the plan.

The party has also struggled to find provisions to raise money to offset major individual and business tax cuts without running into political opposition. Trump on Monday tweeted that "there will be NO change" to the popular 401(k) contribution benefit, a tool considered to help to offset the cuts.

The GOP has also faced opposition to a proposal to get rid of state and local tax deductions. Those provisions affect high-tax blue states the most.

The Senate GOP will have to walk the same tightrope that it did when two high-profile attempts to repeal the Affordable Care Act failed earlier this year. The party can only lose two of its own votes and still pass a plan. Concessions to some lawmakers could risk alienating others.

For instance, while Corker has sounded the alarm about deficits, Sen. Rand Paul, R-Ky., has raised concerns that the proposal does not go far enough to reduce the tax burden on the middle class.

Paul previously cited an analysis from the Tax Policy Center that estimated 28 percent of middle-income Americans could eventually see a tax increase under the plan.

— Reporting by CNBC's Ylan Mui.