Senate Majority Leader Harry Reid, D-Nev., gave a high-profile boost to Sen. Kirsten Gillibrand on Tuesday by supporting her legislation to remove the prosecution of serious crimes in the military from the chain of command.

“I’m going to support Gillibrand, and [Armed Services Chairman Carl] Levin knows that,” Reid told reporters after the Democrats’ weekly caucus luncheon.

But Reid’s backing alone does not appear to be enough to push the New York Democrat’s amendment, which could receive a vote as soon as Wednesday, over the 60-vote threshold. Reid’s position reflects the majority of his rank-and-file Democrats, who prefer Gillibrand’s measure to the Senate Armed Services Committee’s base text. The committee text includes major changes that Gillibrand and her allies say do not go far enough to protect victims of violent crime and sexual assault, or encourage them to report attacks.

It’s unclear whether Reid’s position conflicts with that of President Barack Obama, as White House Press Secretary Jay Carney dodged a chance to weigh in on the issue Tuesday.

As CQ Roll Call reported previously, Gillibrand has been aggressive in her campaign to remove the chain of command from prosecutions, challenging the military establishment and her own party in ways that have rubbed some colleagues the wrong way. The New York Democrat said Tuesday she has the majority of the Senate supporting her bill; she said she believed 20 to 25 senators were up for grabs and vowed to lobby each one of the undecided members after completing a more-than-hourlong media availability in front of the cameras.

Gillibrand’s margins, however, seem smaller than that, sources tracking the bill said. Senate aides believe the legislation will get more than 50 votes and perhaps could garner as many as 57 or 58 backers. And although she spoke Tuesday with some of her most impassioned tones yet, Gillibrand began to hit a few more conciliatory notes as well, taking a moment to praise and tout the changes made to the underlying committee bill. She, of course, was quick to add that she did not believe those changes went far enough. But putting herself in the position to take credit for creating the legislative environment that fostered change if and when her bill fails could give Gillibrand political momentum to continue her fight without alienating more colleagues.

To that end, Sens. Barbara A. Mikulski, D-Md., and Susan Collins, R-Maine, organized a block of speeches from female senators Tuesday morning to decry the crisis of sexual assault in the military, while also speaking about the differences some had over approach.

A worry among Democrats especially was that Gillibrand and her allies, in making their case for her bill, would make opponents of the legislation out to be “anti-victim.”

Sen. Claire McCaskill, D-Mo., outlined this dynamic to CQ Roll Call last week, before later conceding in a news conference that she and the supporters of the Armed Services Committee bill were losing the PR war to Gillibrand.

Sen. Lisa Murkowski, R-Alaska, who supports the Gillibrand legislation, seemed to best convey the message Mikulski and Collins had hoped would be made on the floor by the handful of women who rose to speak.

“My purpose this morning in joining with my female colleagues here in the Senate is not to argue for or against one amendment or another, it’s to point out that the [National Defense Authorization Act], as reported by the Armed Services Committee, includes so many provisions agreeable to all that truly have a positive impact going forward,” Murkowski said.

Female senators have been debating how to address sexual assault in the military both publicly and behind closed doors since last summer, when Mikulski called for a private meeting to discuss the topic.

The debate over the competing measures comes as Reid looks to finish the defense bill this week. Reid already has filled the amendment tree, and Levin said the Senate needs to wrap up work in the coming days to have enough time to iron out differences with the House.

“I don’t think we have any choice. I think we have to do it before Thanksgiving or else we can’t get it done. We won’t be able to go to conference,” the chairman said.