Decades ago there was a hit song in India, “Chitthi Aayi Hai,” a tear-jerking anthem about mothers mourning the loss of sons who went to school abroad and never came back.

The tune has changed. “It used to be by default students from India and China, in particular, bought one-way tickets over here,” said Vivek Wadhwa, who has written extensively on international tech talent. “Now it’s a two-way street. They work for a few years and go home.”

Mr. Wadhwa, a fellow at the Rock Center for Corporate Governance at Stanford, suggests a reason: After earning their graduate degrees in the United States, potential entrepreneurs and programmers are unable to get work visas.

According to a Brookings Institution analysis, in 2010 only 30 percent of foreign graduates of U.S. universities received an H-1B visa, awarded to temporary workers in specialty occupations. And though 64 percent of all international students in science, technology, engineering and math, or STEM, come from India and China, they are still subject to the 7 percent cap per country that immigration law places on green cards. The wait is so long for permanent residency, many simply leave.

In what they believe will stimulate innovation and the economy, tech companies, lawmakers and universities have been urging the administration to make it easier for international students to stay in the United States. Thus, the fate of foreign graduates has become yet another spark in the contentious debate over immigration reform, prompting a variation on a familiar question: Do foreign graduates get jobs at the expense of American workers?

That is what a technology workers union out of Washington State argued when it sued the Department of Homeland Security last year over an extension to its Optional Practical Training program. OPT allows international students to work for one year in the United States; in 2008, in response to the recession, the department had granted STEM graduates an additional 17 months. But the Federal District Court hearing the suit has ruled that the department had erred by skipping the formal approval process and has until February to present its plan for public review, and then offer a revised rule. Meanwhile, the union, the Washington Alliance of Technology Workers, has filed an appeal challenging the legality of the department even making such a policy.

The OPT extension, opponents argue, represents an end-around to the H-1B visa program that makes it easier for companies to hire low-cost foreign labor and for some unaccredited institutions to bring in students for short-term work instead of education. “I would hope that D.H.S. would say this is a really bad idea and we should phase this out,” said John Miano, the lawyer representing the union. “When people are on student visas serving as guest workers, how absurd is that? They are supposed to be students.”

Ostensibly, an OPT extension improves the long odds of being selected in the H-1B lottery, one of the paths to permanent residency. “A longer OPT might mean they get two bites of that apple,” said Ben A. Rissing, a professor at Cornell’s School of Industrial and Labor Relations. But chances are still slim. Congress caps H-1B visas yearly at 65,000, with another 20,000 slots for graduates with advanced degrees. Consider the crush every April: Last spring, there were 233,000 petitions for H-1B visas within the first five days of their being offered.

Amid pleas from Microsoft and other tech companies, two bills were introduced in Congress this year that would eliminate caps on employment-based visas and ease residency restrictions for those with graduate degrees — the “green cards for grads” solution. Both bills are stalled in committees.

In a letter supporting the Senate bill, 14 associations representing thousands of higher education institutions described an urgent need: “According to projections, the United States will face a shortfall of more than 200,000 advanced-degree STEM workers by 2018,” the associations wrote. “In many STEM areas, foreign students are a majority of the Ph.D. graduates from U.S. universities.” It went on to say that the “immigration system forces many of them to leave, sacrificing the innovation and economic growth they would create here.”

Or, as one prominent immigration lawyer, Stephen Yale-Loehr, put it: “Imagine if the next Google or Facebook were to be developed in India or China. All those jobs that could have been in the United States instead are being developed overseas and competing against our best companies.”

For labor proponents, that is far too simplistic. “Basing immigration policy on something as unpredictable as entrepreneurship is one of the things that sounds good,” Mr. Miano said, “but it just doesn’t make sense.”

For graduates working under OPT, the uncertainty is frightening. “You’re gambling your whole career based on some lottery,” said Venkatesh, 27, who did not want his last name published for fear of its affecting job prospects. Having graduated from New York University in 2014 with a master’s in computer science, he now works as a programmer for a large financial institution.

Venkatesh, who is from Hyderabad, India, says that 12 months is not enough time to pay back university loans or to gain sufficient work experience.

“I would probably prefer to find a way to do something better in India,” he said, “because you can’t rely on a broken immigration system.”