The Senate voted today to take up a provision from President Barack Obama’s jobs package that it blocked last month on another procedural vote.

The measure is expected to become the vehicle for another of Obama’s jobs provisions that would help veterans find work, and Democratic and Republican leadership aides said it stands a decent chance of passing the Senate. Senate Democratic leaders hope that the bill becomes the first piece of Obama’s jobs plan to be signed into law.

“As Veterans Day approaches, I urge my Republican colleagues to abandon partisanship and help us honor a commitment to this country’s heroes,” Senate Majority Leader Harry Reid (D-Nev.) said on the floor today.

The legislation would repeal the tax provision that will require federal, state and local governments to withhold 3 percent of nearly all of their contract payments beginning in 2013. The withholding mandate is intended to go after scofflaw government contractors who avoid paying taxes, but supporters of the repeal argue that it would impose significant financial burdens on both the public and private sectors.

A similar measure offered by Senate Republicans fell three votes short of advancing Oct. 21 because of Democratic opposition to its method for offsetting the lost tax revenue. The earlier bill would have rescinded $30 billion from fiscal 2012 discretionary spending, while today’s bill would pay for the $11 billion cost in lost tax revenue by closing a loophole in the health care overhaul law. Republicans argue that the loophole would allow some middle-class Americans to qualify for Medicaid.

The Senate voted 94 to 1 to take up the bill, which is similar to a measure the House overwhelmingly passed 405-16 on Oct. 27. Senate Democratic leaders intend to work with Republicans on a list of amendments to the bill, and it could get a vote on final passage as soon as Thursday if Senate leaders agree to do so.

Democrats are pushing to attach an amendment that would provide tax incentives for hiring unemployed veterans, similar to a provision in Obama’s jobs plan, and job-training initiatives for unemployed veterans. The cost of the amendment would be offset by the extension of fees set to expire as part of a veterans mortgage program.

The amendment would provide a tax credit of up to $5,600 for hiring veterans who have been looking for a job for more than six months, as well as a $2,400 credit for veterans who have been unemployed for more than four weeks but less than six months. The amendment would also provide a tax credit of up to $9,600 for hiring veterans with service-connected disabilities who have been looking for a job for more than six months.

The Democratic proposal would boost education and job-training opportunities for veterans, including providing nearly 100,000 unemployed veterans with up to one year of additional Montgomery GI benefits to train for high-demand jobs at community colleges or technical schools.

The offset for the amendment was originally designed to pay for just the jobs-training legislation, but Senate Veterans’ Affairs Chairwoman Patty Murray (D-Wash.) and House Veterans’ Affairs Chairman Jeff Miller (R-Fla.) announced a deal today to include the tax incentives.

Senate Minority Leader Mitch McConnell said on the Senate floor today that there are other bills that the two parties could be working on, including 15 House-passed bills that the GOP argues would create jobs. The Kentucky Republican’s statement was a continuation of a coordinated strategy with House GOP leaders to push for action on the bills.