Lofgren added that she felt "cautiously optimistic" about Congress holding a vote on broad immigration reform legislation in 2013.

President Obama has said he plans to make immigration reform one of the first legislative priorities he will tackle in his second term in office. Although both Democrats and Republicans have suggested a willingness to pass comprehensive immigration reform legislation, a compromise appears far off.

On Tuesday Sens. Jon Kyl (R-Ariz.) and Kay Bailey Hutchison (R-Texas) introduced a Republican alternative to the DREAM Act, an immigration legislation popular among Democrats. One of the biggest sticking points in the GOP alternative, the Achieve Act, is that it lacks a pathway to citizenship for immigrants to who have lived in the United States for some time.

Diaz-Balart said it was important for legislators to pass legislation that does more than just tinker with immigration.

"Piecemeal parts are better than nothing but [don't] solve the overall problem," said the Florida Republican.

Diaz-Balart said any big immigration bill would likely have to be passed in 2013, ahead of the 2014 midterm elections.

"We have a very narrow window of opportunity for that to happen," said Diaz-Balart.