After two pieces of their jobs bill failed in the Senate because Republicans unanimously opposed offsetting their cost with a tax on millionaires, Senate Democrats have picked another piece of the jobs bill without including such a tax.

Next week, Senate Democrats said, they will vote on the “Vow to Hire Heroes Act of 2011,” which would provide incentives to employers for making the hiring of veterans a priority.

The bill would be added as an amendment to a bill that the House of Representatives passed last week — the 3 percent withholding bill.  That House-passed bill would repeal the imposition of 3 percent withholding on certain payments made to vendors by government entities such as federal government contractors.

The Vow to Hire Heroes Act of 2011, rather than being paid for by a controversial surtax on Americans making more than $1 million a year as the other two failed bills would have, would be covered by existing funding mechanisms within the Department of Veterans Affairs. Democrats say that such a technique has been used by previous Congresses.

“We expect it to pass,” Sen. Debbie Stabenow, D-Mich., said today on a conference call with reporters. “It should pass, given the history of this and the bipartisan support in the House.”

The measure would provide a tax credit to companies of up to $5,600 for hiring veterans who have been looking for a job for more than six months and a $2,400 credit for veterans who have been unemployed for more than four weeks but less than six months. If a company hires a veteran with service-connected disabilities who has been looking for a job for more than six months, the company could receive up to a tax credit of $9,600.

Additionally, the legislation would ensure that all outgoing veterans receive the job training skills they need to find a job and transfer into civilian society. A Transition Assistance Program, a workshop, would be mandatory for service members moving from service to civilian life -providing help with resume writing and career counseling.

 The measure would also help older veterans of past wars by providing 100,000 unemployed veterans up to a year of additional GI benefits that could be used on expanded education or training programs.

 ”There are 14 million veterans who are out there trying to find work every day and it’s time that we can do something for them,” Sen. Patty Murray, D-Wash., said today. “Veterans are not a partisan issue. They are something we can all get behind.”

Democrats believe the bill stands a much better chance of passing in the Senate, likely because of the change in the way the bill is paid for. But with Veteran’s Day coming up, the Democrats also may be keying into the symbolism of a vote so close to the holiday.