WASHINGTON (CNN) -- Senate Majority Leader Harry Reid, D-Nevada, shrugged off a Republican threat to filibuster a Democratic-sponsored resolution opposing President Bush's plan to send more U.S. troops to Iraq, predicting Democrats have the votes to cut off such a move.

"Yes, we do," Reid told reporters. "By my last count, 12 Republicans publicly announced they don't support the surge."

The Democrats -- who currently hold a slim 51-49 advantage in the Senate, which includes Democratic Sen. Tim Johnson, who is currently hospitalized while recovering from brain surgery -- would need 60 votes to stymie a GOP filibuster attempt.

If Reid's count is correct, 12 Republicans would likely get Democrats to the magic number. But the outcome cannot be easily determined independently, in part because not all senators have announced if they support or oppose the troop increase.

Notably, Connecticut Sen. Joe Lieberman, who calls himself an "independent Democrat," voiced support for the Bush plan.