"I just don't think this is the right time to take up this issue, with the border security problems, the drug wars going on across the border, 10 percent unemployment," Senate Minority Leader Mitch McConnell, R-Ky., said on "Fox News Sunday."
The push for immigration reform by the White House and congressional Democrats comes amid widespread protests and threatened legal challenges over the nation's toughest law targeting illegal immigrants, signed Friday by Arizona Gov. Jan Brewer.
Sen. Saxby Chambliss, R-Ga., said on CNN's "State of the Union" other issues pending in Congress must take precedence over immigration reform.
"We've got a lot of work left on our plate between now and the end of the summer, and we're starting on financial regulatory reform," he said.
Calling for tighter border security, Chambliss said, "Until you secure the border, trying to really have an overall reform package on immigration just simply can't be done … . Border states have unique problems when it comes to immigration."
Referring to the Arizona law, he said, "This is one situation where the state of Arizona has decided to take matters into their own hands. And if that's what the people of Arizona want to do, then certainly they have that right."
The law makes failure to have immigration documents a misdemeanor. Law enforcement officers with a "reasonable suspicion" a person is an undocumented immigrant will have the authority to ask about immigration status and arrest people who cannot immediately prove they are in the United States legally.
Sen. Robert Menendez, D-N.J., head of the Democratic Senatorial Campaign Committee, told CNN Congress could and should deal with financial reform and take on immigration and climate change this session.
Menendez said the Arizona law demonstrates the need for immigration reform.
"It is fundamentally wrong to be a second-class citizen just because you have a certain accent or you look a certain way," he said. "That's what Arizona is pursuing."
Reform would secure the borders and provide a pathway to permanent residence, he said.
Celebrity Couples of 2014 [PHOTOS]