Senate Democrats appear to finally have found a formula for passing jobs bills — drop the tax increases. But beyond this week’s plan for the veterans’ jobs package, they aren’t swearing them off just yet.

On Monday, a deal appeared to be at hand to graft a bipartisan veterans’ package onto another bipartisan bill to repeal the 3 percent withholding requirement for government contractors. All of it would be paid for by cuts elsewhere in the budget instead of by the millionaire tax that has been the Democrats’ favorite offset to date.

That doesn’t mean they won’t return to the politically charged tax issue in future weeks as other pieces of the jobs package come up, including a payroll tax cut for the middle class and businesses, senior Senate and House Democratic aides said Monday.

“Americans support the wealthy paying their fair share, and Democrats won’t be backing off that,” one House aide said.

But they are when it comes to veterans.

A Senate Republican leadership aide said that if Democrats go back to using the millionaire tax offset, they would be exposed as not seriously working to pass legislation but instead focused on setting up message votes to score political points.

“They can see results when they offer a truly bipartisan proposal,” said the aide, who noted there was a decent chance the bill would pass the Senate.

“A future effort to add a massive tax hike to bipartisan legislation will be seen for what it’s been all along: a bill designed to fail,” said another GOP aide.

But the Senate Democratic aide said that aside from other parts of the president’s jobs bill — such as extending unemployment insurance — Senate Democrats last summer developed their own list of job-creation legislation that they hope to pass, and some pieces of it will also need offsets.

One measure from that list was legislation passed last month by the Senate to crack down on currency manipulation by China. The House is not expected to take up the bill over fears of starting a trade war.

The Senate defeated two Democratic efforts in recent weeks to pass a proposal to provide $35 billion for teachers and first responders and $60 billion for infrastructure projects. Both are pieces of President Barack Obama’s jobs bill, and both were offset with new taxes on millionaires. The Senate voted Monday to take up legislation on a bill that would repeal the tax provision that mandates federal, state and local governments to withhold 3 percent of nearly all of their contract payments beginning in 2013.

The bill is similar to a House measure approved last month. The legislation’s $11 billion cost in lost tax revenue would be paid for by closing a loophole in the health care law that Republicans argue allow some middle-class Americans to qualify for Medicaid. 

The measure could get a vote on final passage as soon as Thursday if an agreement is reached to do so.

In the meantime, Senate Democratic leaders are pushing to attach an amendment designed to encourage businesses to hire veterans. It would provide tax incentives for hiring unemployed veterans, similar to a provision in Obama’s jobs plan, as well as 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.

Senate Democratic leaders hope Republicans will support the bill because of the repeal of the 3 percent withholding proposal and because the veterans’ amendment is supported by House Veterans’ Affairs Chairman Jeff Miller (R-Fla.). The underlying withholding bill overwhelmingly passed the House, 405-16, on Oct. 27. 

Senate Republicans sought to have their chamber also take up the withholding bill last month, but the proposal did not win the 60 votes needed to overcome a filibuster, in part because the measure would have been offset by a rescission of fiscal 2012 discretionary spending.

Senate Democratic leaders also hope that the measure will be the vehicle to get the first piece of Obama’s jobs plan into law.

The offset for the amendment was originally designed to pay for just veterans’ jobs-training legislation, but an agreement was reached between Miller and Senate Veterans’ Affairs Chairman Patty Murray (D-Wash.) to include the tax incentives.

“This agreement is a bipartisan and comprehensive approach to getting our nation’s veterans back to work,” Murray said in a statement Monday. “It includes Republican and Democratic ideas because getting our veterans the financial security and dignity a job provides should never be partisan. For too long in this country, we have patted our veterans on the back for their service and then pushed them out into the job market alone.”

“Today, we are putting aside politics and putting America’s veterans first. This is how the process should work,” Miller said in a statement. “As in any negotiation, neither party gets everything they want, but we found common ground as the House pledged to do in September. I am hopeful that through the VOW to Hire Heroes Act, we can break the logjam of legislation that is currently pending and get all unemployed Americans back to work.”