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 LeeOvernight Defense: Senate passes 0B defense bill | 3,000 US troops heading to Afghanistan | Two more Navy officials fired over ship collisions Senate passes 0B defense bill Overnight Finance: CBO to release limited analysis of ObamaCare repeal bill | DOJ investigates Equifax stock sales | House weighs tougher rules for banks dealing with North Korea 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.

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Lee's bill is cosponsored by Sens. John CornynJohn CornynNew GOP ObamaCare repeal bill gains momentum Week ahead: Senators near deal to stabilize ObamaCare markets GOP eying 'blue slip' break to help Trump fill the courts MORE (R-Texas), Jim DeMint (R-S.C.), Ron JohnsonRonald (Ron) Harold JohnsonSenate Dems hold floor talk-a-thon against latest ObamaCare repeal bill Sen. Cassidy plans to bring down Medicaid Senate committee schedules hearing on health care block grants MORE (R-Wis.), James Risch (R-Idaho), Marco RubioMarco Antonio RubioOvernight Defense: Senate passes 0B defense bill | 3,000 US troops heading to Afghanistan | Two more Navy officials fired over ship collisions Senate passes 0B defense bill Trump bets base will stick with him on immigration MORE (R-Fla.), and David VitterDavid VitterYou're fired! Why it's time to ditch the Fed's community banker seat Overnight Energy: Trump set to propose sharp cuts to EPA, energy spending Former La. official tapped as lead offshore drilling regulator MORE (R-La.).

Senate Democrats are still sorting out how to proceed with a bill from Senate Majority Leader Harry ReidHarry ReidThe Memo: Trump pulls off a stone-cold stunner The Memo: Ending DACA a risky move for Trump Manchin pressed from both sides in reelection fight 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.