The first battle in the newest round of the judicial wars is intensifying today — and is on course to climax next week, when the Senate will decide whether to fill even one of the three vacancies on what’s considered the second most important federal bench in the nation, the District of Columbia Circuit Court of Appeals.

Senate Judiciary Chairman Patrick J. Leahy, D-Vt., and three fellow Democrats on the panel went before the cameras Tuesday morning to tout the virtues of Patricia Millett, a prominent Washington appeals litigator. She is close, but still shy of securing the 60 votes she’ll need to overcome a filibuster by most Republicans, who assert there is not enough work for the court to justify employing more than the current eight among its 11 authorized judgeships.

It’s also the case that the court is now evenly split between presidential choices from each party, so adding judges nominated by President Barack Obama would likely push the ideological mean to the left.

The cloture vote on Millett will come last in a series of six such roll calls arranged Monday evening by Majority Leader Harry Reid. That gives her allies time to search for the votes they need. After New Jersey’s Cory Booker is sworn in Thursday, Millett looks certain to get support from all 55 senators in the Democratic caucus.

Proponents are hoping to find the rest from Republican women and defense hawks. That’s because Millett would be something of a trailblazer on two fronts: She has argued 32 cases before the Supreme Court, more than all but one other women, and she would become by far the most prominent longtime military spouse in the judiciary. Her husband, Robert King, spent 22 years in the Navy before retiring last year — nine years on active duty and 13 as a reservist, with a stint overseas during the Iraq War.

A group of military spouses who are lawyers have been lobbying the Senate on her behalf this week, focusing particular attention on New Hampshire’s Kelly Ayotte (whose husband was an Air Force pilot for a decade), Mark S. Kirk of Illinois (a Navy reservist) and Lindsey Graham of South Carolina (an Air Force reservist).

Two other GOP senators, Lisa Murkowski of Alaska and Susan Collins of Maine, have often voted to advance nominees they eventually oppose, on the grounds that the filibuster is overused.

A top Senate Democratic aide who requested anonymity in order to speak candidly predicted “a big fight” once the D.C. Circuit nominees hit the floor. But the “word is that we and [the White House] are ready to fight hard.” Reid has said that Democrats are focused on getting at least one more judge confirmed to the court, and has hinted at changes to Senate rules if Republicans stage a filibuster.

President Barack Obama and his Democratic allies have signaled that they are concentrating their efforts on getting only one seat on the D.C. Circuit filled this year, and they have concluded Millett holds the best prospects for confirmation.

The president nominated her and two others in June. But Georgetown professor Nina Pillard has run into stiff opposition because of some of her writings about abortion rights. And Robert Wilkins, a federal trial judge in Washington, will not even advance through the Judiciary Committee before Thursday.