Sessions took a swipe at the I-Squared bill, which he says represents industry interests, and "would further drive down wages for American workers."
Rubio has not explained what constitutes abuse in the H-1B program. Some say the replacement of U.S. workers with temporary visa workers is not an abuse, but a feature of the program that's allowed by law. Cruz joins Donald Trump, who is also seeking the Republican nomination, in campaigning on a specific H-1B reform plan.
The Cruz and Sessions bill joins two other H-1B reform bills, one from Chuck Grassley (R-Iowa) and Dick Durbin (D-Ill.), and another from Sen. Bill Nelson.
Cruz's and Sessions' bill requires an employer to commit to paying the foreign workers either what an American worker who did identical or similar work made two years prior to the recruiting effort or $110,000 -- whichever is higher.
During the debate on comprehensive immigration reform in the Senate in 2013, Cruz had sought a 500% increase in the base H-1B cap, from 65,000 to 325,000. This new bill doesn't address the visa cap, with either an increase or decrease, but neither does the bill by Grassley and Durbin.
For its part, the Nelson bill does seek a reduction in the visa cap, and an increase in wages as well. It distributes the visas to employers based on how much they are willing to pay these workers.
"This bill is a bold step to fix the very broken H-1B program," said Ron Hira, an associate professor of public policy at Howard University, of the Cruz bill. Hira has testified in Congress on the H-1B visa.
The Cruz and Session bill "eliminates the temptation for employers to replace American workers with H-1Bs. It does this by setting a realistic wage floor, one that is equivalent to the wages earned by the replaced Southern California Edison and Disney workers," said Hira.
Daniel Costa, director of immigration law and policy research at the Economic Policy Institute (EPI), said the wage rule is a "marked improvement over the status quo."
But Costa thought the wage rule "could have been slightly more artfully crafted to protect the higher-earning H-1B workers."
For instance, "a software developer in the Silicon Valley earning the average wage would earn $142,376 -- and the entry level wage is close to $100,000. So if a Silicon Valley tech firm hires an experienced software programmer at $110,00 they'd be getting a bargain," said Costa.
Costa said Cruz and Sessions "deserve a lot of credit for the subsections of the bill that would go a long way to protect foreign workers from being exploited, underpaid and in some cases victims of labor trafficking."
Sign up for MIS Asia eNewsletters.