For the first time since the financial crisis, U.S. employers are expected within days to reach a limit on the yearly allotment of applications for coveted skilled-worker visas, a sign of the strengthening economy that means some employers will rely on a lottery to fill key positions.

U.S. companies each year can sponsor a total of 65,000 foreigners with at least a bachelor's degree for a so-called H-1B visa, many of which go to programmers and other specialized workers. The program allocates an additional 20,000 visas each...