WASHINGTON — Both parties are using a brief pre-election session of Congress to make campaign appeals to returning veterans.

Senate Democrats are pushing President Barack Obama’s proposed $1 billion Veterans Jobs Corps to relieve high unemployment among servicemen and women returning from Iraq and Afghanistan. House Republicans are pushing a bill making it a crime to benefit from lying about military services or awards.

Lawmakers in both parties agreed this week to come up with more money to help the Veterans Administration reduce a disability claims backlog.

Obama proposed a job corps for veterans last February that would place them in jobs restoring public lands and beefing up local police and fire departments. It cleared a preliminary test vote Tuesday, as was expected, but aides said progress could easily unravel as lawmakers negotiate what amendments and how many of them will be allowed.

Although the legislation is not expected to become law this session, it gives lawmakers a chance to display their support for the nation’s 21 million-plus veterans before Congress adjourns for the campaign season.

The unemployment rate for veterans of Iraq and Afghanistan has been trending lower in recent months, but hit a bump last month. Joblessness among them was nearly 11 percent in August, compared with an 8.1 percent jobless rate nationwide.

Sen. Bill Nelson, R-Fla., said the problem is likely to grow as troops are withdrawn from Afghanistan.

“For the troops, when they come home, the fight is not over. There’s another fight when they get back home to America,” said Nelson, the bill’s lead sponsor who is in a tough re-election contest. “We need to give them as many opportunities as possible to succeed when they get back home here in America.”

Nelson said Tuesday he still expects the GOP to try to block the bill.

The White House released a statement Tuesday declaring its support for the measure. “Many of these veterans returning home have been unable to find suitable employment commensurate with their training and experience from over ten years of war,” said a statement from the White House Office of Management and Budget. “Furthermore, America’s public lands face enormous unmet infrastructure and facility maintenance needs, and this country’s veterans have the skills to meet these demands.”

House Republicans plan a vote this week on their “stolen valor” bill. The Supreme Court struck down a similar law in June, saying false claims about being awarded medals for military service are “contemptible” but are still protected by the First Amendment.

Rep. Joe Heck, a Nevada Republican in a tough re-election contest, said his legislation is constitutional because it would narrow the scope of the law to punishing just those who seek to financially benefit by lying about their service and awards.

Lawmakers want to return to their districts to campaign for re-election as soon as possible. House members could leave as early as Friday and are expected to stay in Washington no later than the end of next week. The Senate is likely to have a shortened September schedule too.