Wednesday's Senate vote to negotiate a tax bill with the House started the clock on an ambitious plan by Republicans to finalize a sweeping tax overhaul in just a matter of weeks, in the hopes of putting the once-in-a-generation bill on President Trump's desk by Christmas.

Even before the Senate vote, Republican leaders were talking behind the scenes to mull some of the major revisions needed to bring the bills into alignment. Some of the biggest changes to the Senate bill were added just before it passed to expedite the process.

“Before Christmas,” said Senate Majority Leader Mitch McConnell, R-Ky., Wednesday, when asked by radio host Hugh Hewitt about timing for final passage. “We’ll have to have a conference with the House yet ahead of us and work out the differences, but the core of the two bills are really very similar. And I think we’ll be able to do that in fairly short order."

Although the version of the bill passed by the Senate created new problems to fix, Republicans projected optimism that the obstacles were not a threat to final passage.

The broadest issues to be addressed are whether the conference committee can maintain all the promises made to members to gain their votes without giving up on the 20 percent corporate tax rate and how to fix the Senate’s last-minute decisions to retain the individual and corporate alternative minimum taxes.

Senate Republicans added the corporate AMT and a reduced individual AMT back into the bill at the last minute on Friday night in order to free up revenues for other tax breaks required by senators to earn their votes. Those inclusions were aimed at advancing the bill with an eye toward correcting it in conference.

The addition of the corporate AMT “is a placeholder, but also it was not put in very artfully,” said Evan Migdail, a partner at DLA Piper. The late addition of the corporate AMT drew criticism from, among others, the U.S. Chamber of Commerce.

The problem is that the bill would lower the corporate tax rate from 35 percent to 20 percent, but it wouldn’t change the AMT rate, which is also 20 percent. As a result, the AMT would be the relevant tax for more companies. And it would negate the value of incentives that Republicans were careful to include in the reform, such as the research and development tax credit and the special low rate on intellectual property income meant to prevent pharmaceutical and tech companies from moving their businesses overseas.

“The scary thing about this is it scrambles all of the policy that is in the current tax bill,” said Phil English, a senior government relations advisor at Arent Fox and former member of the tax-writing Ways and Means Committee.

Key lawmakers, such as conference committee chairman Rep. Kevin Brady of Texas, have said they plan to work toward taking the AMTs out of the final bill.

Migdail suggested that the corporate AMT is likely either to be stripped out in conference or the AMT rate lowered to around 11 percent.

Similarly, the individual AMT added back into the bill would partly undermine its purpose by becoming the relevant tax for a large group of people. The vast majority of rich, but not super-rich, taxpayers would pay the AMT under the Senate bill, according to an analysis published by the Tax Policy Center Wednesday. All married couples with ordinary income between $290,000 and $765,000 would pay the AMT, the think tank found.

Len Burman, a fellow at the Tax Policy Center, called the AMT an “ugly, backdoor way” of raising taxes. “I don’t think it belongs in any reform bill,” he said.

Including the individual AMT in the bill, however, gave Senate tax writers $133 billion in revenues on paper to dedicate to other tax cuts.

Many Republicans want to replace that offset with something else. “A lot of us want to see the AMT gotten rid of,” White House legislative director Marc Short said in an interview with CNBC.

One of the most straightforward ways to recoup the savings would be to ease off the goal of lowering the corporate tax rate to 20 percent. Instead, they could aim for a higher rate, as Trump suggested over the weekend.

On Wednesday, however, Short downplayed that possibility. “We do have the votes to get 20 percent, and that’s what we’re trying to do ... that is a priority for us,” he said.

Asked off the Senate floor Wednesday afternoon if a 22 percent rate could be a possibility, Senate Finance Committee chairman Orrin Hatch, R-Utah, denied it. “Not as far as I’m concerned,” he said. “We’re still at 20.”

Any number of other changes, big and small, could be in the works. Some would involve the House bill becoming more like the Senate version. For example, on Wednesday, House members expressed interest in adding back in deductions for teachers’ out-of-pocket expenditures, student loan interest, and medical costs, all of which are in the Senate version. House Republicans also generally favor doing as the Senate did and repealing the Obamacare individual mandate as part of the bill.

Matching up all the different priorities and the revenues needed to stay within the $1.4 trillion tax cut limit will be tricky. McConnell suggested that obstacles would melt away in proximity to passage. “There are some dials that will have to be twisted, and there are some complaints,” he said. “But there’s overwhelming support among businesses both large and small.”

On Wednesday, Trump sounded eager to get on with the conference. “I call it ‘the mixer,’” he said in brief remarks at the White House. “It's in conference right now, but I call it ‘the mixer.’ I think when it comes out, it's going to be a beautiful mix.”