Doug Jones, the apparent winner of Alabama's Senate election, has signaled that he would oppose the GOP tax plan.
The election pits Republican ex-Judge Roy Moore against Democratic former federal prosecutor Doug Jones.
Sen. Rand Paul, R-Ky., said he "cannot in good conscience vote to add more to the already massive $20 trillion debt."
A Senate version of the tax bill would repeal Obamacare's rule requiring most Americans to have health insurance or pay a fine.
With the vote, the GOP leaps closer to its promise of dramatically remaking U.S. tax law.
A flurry of last-minute changes Friday to the Senate Republican tax bill have set the plan on a fast track to passage.
CNBC's Ylan Mui reports that Tennessee Senator Bob Corker is a no vote on the tax reform package.
Holdout Sen. Bob Corker says he thinks the GOP can approve it even without his vote.
Sens. Johnson and Daines reach an agreement with the GOP leadership to raise the amount of pass-through income business owners can deduct.
Senate Majority Leader Mitch McConnell said the next in a series of roll call votes is set for Friday morning.
Analysis by economists in both parties has shown that no tax cut in modern history has ever "paid for itself" by generating a surge in revenues.
The Senate parliamentarian has ruled that the fiscal trigger will not work under the rules of the Senate.
The Senate tax plan would add more than $1 trillion to federal deficits even after the effects of economic growth.
Substantial changes are still being made to the Senate GOP tax plan — which could be passed by the end of the week, Vox reports.
A tax bill tool designed to win the support of key GOP senators is finding resistance among others.
U.S. Senate Republicans rammed forward a tax-cut bill in an abrupt, partisan committee vote that set up a full vote by the Senate as soon as Thursday.
As they search for ways to join the Republican tax cut push, self-styled "deficit hawk" senators have raised questions that appear to make no sense.
Some GOP senators haven't yet committed to supporting the tax bill, but anti-tax crusader Grover Norquist thinks the legislation will pass.
With the party-line 12-11 vote to advance the plan, Republicans overcame one possible roadblock.
"The ball is in the Senate's court," House Ways and Means Committee Chairman Kevin Brady says.