WASHINGTON — Testing the waters of what is expected to be a turbulent battle over immigration policy next year, the House voted Friday to make green cards accessible to foreign students graduating with advanced science and math degrees from U.S. universities.

But even this limited step, strongly backed by the high-tech industry and enjoying some bipartisan support, is unlikely to go anywhere this session of Congress, dramatizing how difficult it will be to find lasting solutions to the nation’s much-maligned immigration system.

A more sweeping bill presumably would deal not only with legal residents but also the estimated 11 million people here illegally.

Republicans largely shunned by Hispanic voters and other minorities in the November elections used Friday’s 245-139 vote for the STEM Jobs Act to show they have softened their hardline immigration policies and are ready to work for more comprehensive legislation.

Republican leaders added a provision making it easier for immigrants working in the country legally to bring their spouses and children to the United States while they wait for their visa applications to be approved. Typically, family members now wait more than two years to be reunited. About 80,000 such family-based visas are issued every year.

But for many Democrats and the Obama White House, this first step was more of a misstep.

Democrats, including members of the Congressional Hispanic Caucus, assailed the legislation for offsetting the 55,000 new permanent residency visas by eliminating a program that provided green cards to people with traditionally lower rates of immigration, particularly those from Africa. STEM stands for science, technology, engineering and mathematics.

The White House, in a statement opposing the GOP-crafted bill, said it was encouraged that Congress “appears to be ready to begin serious debate on the need to fix our broken immigration system.” But it said the administration does not support “narrowly tailored proposals” that do not meet long-term objectives of achieving comprehensive reform.

That approach includes dealing with the young people brought into the country illegally, establishing a solution for agriculture workers, creating an effective border enforcement system and worker verification program and deciding by what means those living in the country illegally can attain legal status.

The Democratic-controlled Senate is seen as likely to ignore the House STEM bill in the waning days of the current congressional session.