A divided Congress will attempt to shed its “do nothing” moniker by passing a small piece of President Obama’s $447 billion jobs bill this week.

After 10 months of clashes over big bills – repeal of president’s signature health care reform, then gridlock over a new White House jobs bill – leaders on both sides of the aisle are turning to smaller bills with some prospect of bipartisan support.

The Senate this week takes up a House-passed bill to boost job creation by repealing a 3 percent tax on government contractors, set to take effect in 2013. It’s the first House-passed portion of the president's jobs bill to be allowed on the Senate floor for a vote.

In a show of rare bipartisan agreement, the Senate voted on Monday to begin debate on the bill by huge margin, 94 to 1. More than 15 other House bills, mainly focusing on creating jobs by curbing government regulation, are languishing in the Senate, with little prospect of a floor vote.

But with congressional approval ratings now at record lows – in the single digits – neither party wants to head into 2012 elections with a record of near-zero legislative accomplishment, especially on a topic as critical to voters as jobs.

In a nod to Veterans Day this week, Senate Democrats are also proposing a package to help returning veterans find jobs in a tough market environment. The amendment, which draws on work by the veterans affairs committees in both the House and Senate, proposes tax credits for firms that hire veterans, and increases incentives for hiring disabled veterans.

In a another shift in tone, neither majority Republicans in the House or majority Democrats in the Senate insisted on “poison pill” features that derailed previous bids at a jobs bill, such as raising taxes on the rich (a nonstarter for Republicans) or rolling back environmental regulations (unacceptable to Democrats).

The veterans amendment “contains many provisions supported by Republicans,” said Senate Republican Leader Mitch McConnell of Kentucky, in a floor speech on Tuesday. “So maybe we’re making some progress here.”

“There's a lot we can agree on when it comes to jobs legislation,” he added. “That's where the focus should actually be.”

There are also bipartisan calls in the Senate to take up two measures to help small businesses that passed the House last week on near unanimous votes.

One bill cuts costly financial reporting requirements to community banks, expected to expand lending to small businesses. The other makes it easier for small businesses to raise capital by easing registration requirements for small companies going public.

Sens. Jon Tester (D) of Montana and Pat Toomey (R) of Pennsylvania are also sponsoring a similar bill.

This push for common ground marks a clear shift from where Senate Republicans and Democrats have stood on jobs legislation for months. Senate Republicans, voting unanimously, blocked the president’s full $447 billion jobs package and the first two efforts to pass elements of that bill in separate parts.

Both bills – aid to first responders and, later, a plan to boost public investments in infrastructure – were paid for by a surtax on incomes over $1 million, a move Republicans called a “job killer” and refused to back.

Instead, Senate Democrats this week propose paying for repealing the 3 percent contractors’ tax by extending a fee on VA-backed mortgages – a noncontroversial solution backed on both sides of the aisle.

In addition to tax breaks for firms that hire veterans, the measure proposes an additional year of job training and tax credits for veterans with service-related disabilities. It also provides mandatory transitional assistance for returning veterans and allows service members to apply for jobs before they leave the military.

With unemployment rates for veterans at 12.1 percent – that’s more than 3 percent above the overall US unemployment rate – there’s pressure on both parties to pass this measure. Unemployment rates for the youngest, post 9/11, veterans are nearly 22 percent.

In a press briefing on Monday, White House spokesman Jay Carney renewed calls on Congress to act on the jobs front.

“There's great frustration in the country with the dysfunctionality that has been demonstrated in Congress lately, with its refusal to take very simple, bipartisan, mainstream decisions that would have a positive impact on economic growth and job creation,” he said.

“So we welcome every prominent voice who joins in the effort to call on Congress to do its job.