The Hatch/McConnell bill appears to be similar to the bill they introduced last week, S. 3413. Last week, Hatch said Democrats "would prefer to push America's economy off the fiscal cliff" rather than prevent a small business tax increase.

"The American people deserve better than to have the President and his allies threaten to melt down our economy for what amounts to less than three days of federal spending," Hatch said last week, referring to the revenue Democrats are hoping to raise by allowing tax rates to rise on people earning more than $250,000 a year.

"Economic growth is slower today than it was when the President agreed to stop these massive small business tax hikes," Hatch said. "With 41 consecutive months of unemployment over 8 percent, it just makes sense to extend this tax policy for a year so we can enact meaningful, pro-growth tax reform."

The other Utah Republican, Sen. Mike LeeMichael (Mike) Shumway LeeSenators to Trump: Let Mueller finish Russia probe The Hill's Morning Report: Can Trump’s VA pick make it through the week? Mike Lee pens op-ed calling legislation to protect Mueller 'unconstitutional' MORE (R ), introduced a more aggressive proposal on Monday to permanently extend all the tax levels. Lee's bill, S. 3420, would also permanently fix the AMT, and also repeal the estate tax.

Lee's bill is cosponsored by Sens. John CornynJohn CornynGOP moves to cut debate time for Trump nominees Republicans want Trump’s VA nominee to withdraw Senators to Trump: Let Mueller finish Russia probe MORE (R-Texas), Jim DeMint (R-S.C.), Ron JohnsonRonald (Ron) Harold JohnsonGOP moves to cut debate time for Trump nominees Election security dominates hearing for Trump Homeland Security nominee Former officials offer support for Trump Homeland Security nominee MORE (R-Wis.), James Risch (R-Idaho), Marco RubioMarco Antonio RubioLobbying world Former Florida congressmen mull bipartisan gubernatorial run: report Winners and losers from Jim Bridenstine’s confirmation as NASA administrator MORE (R-Fla.), and David VitterDavid Bruce VitterPlanned Parenthood targets judicial nominee over abortion comments Trump nominates wife of ex-Louisiana senator to be federal judge Where is due process in all the sexual harassment allegations? MORE (R-La.).

Senate Democrats are still sorting out how to proceed with a bill from Senate Majority Leader Harry ReidHarry Mason ReidGOP moves to cut debate time for Trump nominees Harry Reid: ‘The less we talk about impeachment, the better off we are’ Lobbying world MORE (D-Nev.), including how many amendments might be allowed. Republicans will then have to decide what amendments to push, but are at least expected to push for Hatch's language providing for a one-year extension of low tax rates.

The Senate could hold a vote on whether to proceed to Reid's bill, S. 3412, on Wednesday.

Republicans this week continued to make the case that no taxpayer should face higher rates next year, given the struggling economy and President Obama's own arguments in years past that taxes should not be increased on anyone during a weak economy. They also accused Obama of trying to pit one class of Americans against the other.

On Monday, Minority Whip Jon Kyl (R-Ariz.) argued that President Obama is making a "misguided and wrong and even dangerous" case that income earners can be grouped into different classes that receive different treatment.

"Each day the President is out on the campaign trial championing himself as the great protector of what he calls the middle class, and pitting those Americans against their fellow citizens by arguing that the wealthiest class is victimizing them through the tax code," Kyl said.

"If wealthy people are not made to pay more, he argues, the middle class will be stuck in their current stations," Kyl said. "What one class wins, he implies, the other class loses. In this, I believe he is wrong. Moreover, I believe such a formulation is contrary to four centuries of American history."

— This story was updated at 10:11 a.m. to add information about the Hatch/McConnell bill.