Companies that want to hire foreign workers with specialized skills and expertise are facing a growing bottleneck with the H-1B visa program, which allows skilled foreign employees a visa if they are sponsored by an American employer.

The number of visas is capped at 65,000 a year, with an additional 20,000 possible under a special program for applicants with master’s degrees and doctorates.

This year, the government was flooded with 124,000 applications, and the cap was reached in less than 24 hours. That means that many thousands of companies that want to offer a foreign worker a job will be unable to do so.

At least one bill in Congress is aiming to solve the problem. Senate Bill 169, which has bipartisan support, would raise the cap to 115,000 a year, with the possibility that it will eventually reach 300,000.

But not everyone likes the idea; labor groups and others have complained that expanding the cap encourages hiring foreign workers at the expense of Americans. A competing bill in the Senate (S. 600) would force companies to demonstrate that they made a good-faith effort to hire an American – including listing positions on a government-sponsored website for 30 days – before they could sponsor a foreign worker for a visa.