Sens. Kay Bailey Hutchison (R-Texas) and Jon Kyl (R-Ariz.), both retiring, introduced the Achieve Act, meant to be an alternative to the DREAM Act supported by Democrats. The Achieve Act would allow illegal immigrants who came to the United States as children to stay in the country, similar to the DREAM Act. Unlike the DREAM Act, the Achieve Act does not offer a pathway to citizenship. 

President Obama has vowed to make immigration reform a top priority of his second term in office. Earlier this year, Obama issued a directive meant to protect immigrants living in the country illegally from being deported, provided they meet certain criteria and came to the United States as children. Flake said that directive was only a temporary fix, a sentiment shared by other Republicans.

"What we have right now with President Obama's action is a temporary measure. We need something permanent," Flake continued. "And this has always been an issue that we have needed to deal with in a humane, rational way. And if this bill represents the place to start, that's great."

Both Republicans and Democrats have suggested strong interest in passing comprehensive immigration reform. Rep. Luis GutierrezLuis GutierrezJudiciary Dem asks GOP chairman to invite Trump to testify in public The Hill's 12:30 Report Dems plan to sue Trump over conflicts of interest: report MORE (D-Ill.), one of the most vocal proponents of passing immigration reform, recently said Sen. Marco RubioMarco RubioWill Republicans stand up to the NRA's insurrection rhetoric? The Hill's Whip List: Senate ObamaCare repeal bill Ivanka Trump turns to House GOP on paid family leave MORE (R-Fla.), a rising star in the GOP who is often mentioned as a possible presidential candidate in 2016, will be key to a bipartisan agreement on immigration.

"He'll play a critical role — a necessary, critical role," Guitierrez said of Rubio.