Proponents of overhauling the U.S. immigration system increasingly point to the fact that about 40% of the 11 million undocumented workers in the country aren't low-wage workers who sneaked over the southern border illegally, but rather foreigners who arrived legally and simply never left.

Those working to create a path to citizenship for people here illegally often make the distinction to highlight the diverse immigration issues the U.S. faces. Little is known about the demographics of the so-called overstayer population, but some studies suggest they tend to be better educated and more fluent in English than those who crossed the border illegally. They also are more likely to hail from European, Asian and African countries. And in many cases, they used tourist visas to enter the U.S.

A bipartisan Senate group, returning this week from its spring recess, is in the final stretch of an effort to produce legislation. But senior lawmakers disagreed on Sunday news shows about whether it would be ready this week. New York Sen. Charles Schumer, a Democrat, said negotiators were on track to reach an agreement by the end of the week. But Sen. Lindsey Graham, a South Carolina Republican, was more cautious, saying it could take a couple of weeks for senators to hash out details on low-skilled guest-worker programs and high-skilled visas.

The Senate effort has been repeatedly delayed, in part because so many facets of immigration are more complex than many realized. Among those issues: the 40% of unauthorized immigrants who overstay their visas.

"Whenever I use that statistic, people seem completely surprised," said Sen. Jeff Flake (R., Ariz.), part of the Senate group working on immigration. "They assume that, yeah, some people overstay—but 40% of the illegal population here?" A good chunk of those, he said, could prove to be people who overstayed H-1B visas, which are given to highly skilled workers.