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 LeeNSA spying program overcomes key Senate hurdle With religious liberty memo, Trump made America free to be faithful again This week: Time running out for Congress to avoid shutdown 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 CornynMcCarthy: ‘No deadline on DACA’ NSA spying program overcomes key Senate hurdle Hoyer suggests Dems won't support spending bill without DACA fix MORE (R-Texas), Jim DeMint (R-S.C.), Ron JohnsonRonald (Ron) Harold JohnsonGOP senators eager for Romney to join them The House needs to help patients from being victimized by antiquated technology Comey’s original Clinton memo released, cites possible violations MORE (R-Wis.), James Risch (R-Idaho), Marco RubioMarco Antonio RubioOvernight Cybersecurity: Bipartisan bill aims to deter election interference | Russian hackers target Senate | House Intel panel subpoenas Bannon | DHS giving 'active defense' cyber tools to private sector Senators unveil bipartisan push to deter future election interference Puerto Rico's children need recovery funds MORE (R-Fla.), and David VitterDavid Bruce VitterWhere is due process in all the sexual harassment allegations? Not the Senate's job to second-guess Alabama voters The Senate 'ethics' committee is a black hole where allegations die MORE (R-La.).

Senate Democrats are still sorting out how to proceed with a bill from Senate Majority Leader Harry ReidHarry Mason ReidDems search for winning playbook Dems face hard choice for State of the Union response The Memo: Immigration battle tests activists’ muscle 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.