Sen. Mike Lee is taking the lead on the issue in the Senate and has offered an identical bill.
Their proposal would eliminate the per-country caps on visas issued to highly skilled workers, essentially allowing more well-educated people from India and China to work in the United States.
Currently each country can claim only 7 percent of the 140,000 work visas issued annually. The bill also would boost the cap from 7 percent to 15 percent on family visas, allowing more people from places such as Mexico to reunite with their families.
The legislation sidesteps the hot topic of illegal immigration. It also doesn't increase the total number of visas provided in a year.
While a wide swath of Democrats and Republicans supports the measure, Grassley has reservations.
"I have concerns about the impact of this bill on future immigration flows," he said, "and am concerned that it does nothing to better protect Americans at home who seek high-skilled jobs during this time of record high unemployment."
Lee and Chaffetz, along with Sen. Marco Rubio, R-Fla., plan to visit soon with Grassley.
"I simply think it is a matter of education and communicating with him," Chaffetz said. "The only way to even get one of these visas is you have to demonstrate that there is no American who has applied to fill that job."
Lee explained the case he plans to make to Grassley.
"It doesn't make a whole lot of sense to have these antiquated quotas in place and a country-by-country basis," Lee said. "Tell me how that does anything to help the American worker? I don't know that it does."