WASHINGTON — Former congresswoman Gabrielle Giffords is returning to Capitol Hill on Wednesday to help kick-start a longshot campaign to expand criminal background checks to all commercial firearms sales.

Similar legislation that sought to expand background checks failed to get a hearing in the House last session. With the GOP expanding its majority and winning control of the Senate, prospects for the bill may be even more unlikely this session.

Still, Democratic Rep. Mike Thompson of California predicted the legislation would pass if GOP leadership would let it come to the House floor for a vote. Congress passed record funding for background checks in the last session, he said.

“If they are willing to fund the system at historic levels, they should support using the system,” Thompson said.

Under the current system, cashiers at stores selling guns call in to check with the FBI or other designated agencies to ensure the customer doesn’t have a criminal background. Many lawmakers want to expand such checks to sales at gun shows and purchases made through the Internet.

The National Rifle Association opposes expanding background checks. The organization says many people sent to prison because of gun crimes get their guns through theft or the black market.

Giffords co-founded Americans for Responsible Solutions. The organization advocates for stricter gun laws. Giffords is scheduled to be at the news conference with Thompson and a handful of other lawmakers on Wednesday. The Arizona Democrat has become an increasingly active player in the gun-control movement since being shot in the head as she met with constituents in Tucson nearly four years ago.

The legislation that ushered in background checks for guns bought from federal licensed dealers was named after James Brady, the press secretary to President Ronald Reagan who was shot in the head in 1981 and died last year.

“We fought a long, hard battle to pass the Brady Bill with bipartisan support in 1993 and now we simply need to finish the job!” said Sarah Brady, co-founder of the Brady Campaign to Prevent Gun Violence.