The Senate Finance Committee rejected two amendments on Tuesday that would have put a public health-insurance option into that committee’s health-system reform proposal.

The first amendment, introduced by Sen. Jay Rockefeller (D-W.Va.), was voted down 15-8.

Committee Chairman Max Baucus (D-Mont.) said he voted no on the measure because he didn’t think the bill could pass the Senate with a public option attached.

“Now, I can count.  And nobody has been able to show me how we can count up to 60 votes with a public option in the bill -- and thus, I'm constrained to vote against the amendment,” Baucus said. “ My larger goal is to enact healthcare reform.  I want the strongest bill I can possibly get.  I want a bill that will become law.”

The second amendment, introduced by Sen. Charles Schumer (D-N.Y.), failed by a vote of 13-10.

Senate Majority Leader Harry Reid will introduce a final Senate bill without a pubic option, according to Senate Democratic leadership sources.  While supporters may try to introduce a public option amendment to the eventual Senate floor bill, it would likely be defeated or denied a vote.

If the Senate and House pass separate bills, lawmakers would reconcile their differences in a conference committee.  There, negotiators could  insert the government-run option.  If so, and supporters can persuade a handful of their conservative Democrat colleagues to vote their way, the Senate would have a last shot to approve a bill with a public plan. 

The public option, according to recent polls, has tepid support with the American public. It has virtually zero support with Republicans on Capitol Hill, and even some more-conservative Democrats are opposed to it.

After the vote, Schumer acknowledged that a bill with a public option cannot pass the Senate at this time, but he said he hopes to change his colleagues’ minds.

However, since the Senate Health, Education, Labor and Pensions Committee as well as several House committees have passed bills that do include a public option, reconciling the different versions could become more difficult.

The public option is controversial because some say that a government-run option would keep health-insurance companies from taking advantage of new insurance rules to profit excessively; but the insurance industry and others argue that the government option could offer prices that are too low and wouldn’t be fair competition for private-sector offerings.

Health insurers such as WellPoint (WLP), UnitedHealth Group (UNH) and Cigna (CI) saw their stocks retreat during Tuesday’s session.

--FOX News's Trish Turner contributed to this article.