This post has been updated.

The Senate is set to take up a bipartisan bill Tuesday addressing the Medicare "doc fix," a yearly legislative headache on Capitol Hill, but some of the Senate's most conservative members are threatening to oppose the legislation crafted last month in the House, citing its price tag.

Majority Leader Mitch McConnell (R-Ky.) said Tuesday that he expected to bring the bill to the Senate floor later in the day, hours before Medicaid officials say they will implement a 21 percent cut to the rates it pays doctors treating Medicare patients. McConnell said there is "a bipartisan desire" to pass the bill before the midnight deadline.

Complicating passage is the opposition of several staunch Senate conservatives, including presidential candidate Ted Cruz (R-Tex.), who said in a statement Tuesday that any deal ending the "doc fix" should be "fully paid for and include significant and structural reforms to Medicare that provide seniors more power and control over their health care."

The Congressional Budget Office estimates that the House-passed bill would add $141 billion to federal budget deficits through 2025. In the subsequent 10 years, the nonpartisan budget analysts said, "federal savings would increase rapidly" but they declined to quantify to what degree those those savings would cut into projected deficits, citing "the considerable uncertainty that exists about the evolution of the health care delivery and financing systems." One outside analyst, the Committee for a Responsible Federal Budget, suggests the total cost over 20 years could approach a half-trillion dollars.

House Republican leaders have played down the price tag and touted Medicare reform provisions in the legislation that would "bend the cost curve" in the long term. The bill, forged through high-level bargaining between Speaker John A. Boehner (R-Ohio) and Minority Leader Nancy Pelosi (D-Calif.), passed 392 to 37 on March 26.

But Sen. Jeff Sessions (R-Ala.) said Monday he would insist on fully offsetting the $141 billion in net spending with other federal budget cuts. "I'm disappointed, frankly, that the House sent over a bill that was not paid for," he said. "It's not easy, but it's very doable."

Beyond the cost, Cruz said the legislation also does nothing to address his signature issue, dismantling the Affordable Care Act, and instead "institutionalizes and expands Obamacare policies that harm patients and their doctors while adding roughly half a trillion dollars to our long-term debt within two decades."

Majority Whip John Cornyn (R-Texas) said Tuesday he had filed an amendment that would pay for the bill by ending the Affordable Care Act's individual mandate -- a change likely to win support from virtually all 54 Senate Republicans and perhaps a few moderate Democrats. But it was unclear Tuesday afternoon whether such an amendment would have to win a simple majority or clear a higher 60-vote threshold.

Any successful amendment would mean the bill would have to go back to the House -- thus busting the Medicare deadline. "Honestly, it's my hope that the amendments are not approved, because we need to get this bill down to the president for signature before midnight," McConnell acknowledged Tuesday.

Asked about the bill's cost, McConnell said, "Like any bill of this magnitude, there are strengths and weaknesses."

McConnell said he was particularly pleased that the bill includes "means testing" limiting benefits for wealthier Medicare recipients. "I think that issue alone commends the bill," he said. "Like any large bill, it's a mixed bag in some respects, but I think on the whole it's a bill well worth supporting and we also need to pass it today, and I think most of our members understand that."

Senate Minority Leader Harry M. Reid (D-Nev.) said Tuesday that Democrats would offer amendments of their own, including one addressing abortion language in the bill and another further extending a two-year reauthorization of the Children's Health Insurance Program.

Sessions admitted Monday that there may not be enough opposition to the additional spending to derail the legislation: "On a really important need like the physicians' payments, if the leadership insists that the bill go forward without a pay-for with Democrat votes, there will be enough Republican votes to pass it."