Democratic senators on Sunday called for Congress to promptly take up comprehensive immigration legislation, saying the controversial policy signed into law in Arizona this past week highlights the need for a national overhaul.

Republicans insist the time is not right for the legislation, with unemployment near 10 percent and negotiations over a Wall Street regulation bill in full swing. Democrats are already seeing their work on a climate change bill fall apart as a result of shifting focus to immigration.

But Democrats are defending the decision to pivot, rebutting charges that they're doing so to woo Hispanic voters ahead of the midterm elections. Sen. Christopher Dodd, D-Conn., said Sunday that Congress needs to act to make sure other states don't follow Arizona's lead.

"The idea that state by state would start developing its own immigration laws in the country -- imagine what a patchwork that might look like," Dodd told NBC's "Meet the Press." "It's demanding a national answer to immigration policy, so before this even gets further out of hand, we've got to step up and do the job."

Dodd called the Arizona law "outrageous."

He's not alone. Hundreds of people were planning to protest the law in Phoenix Sunday afternoon. President Obama has decried the policy as "misguided" and called on the Justice Department to look at its civil rights implications.

The law would make it a state crime to be an illegal immigrant and require police to question individuals about their immigration status if they suspect they could be in the country illegally.

But a recent Rasmussen poll found that despite the outcry nationally, Arizonans generally support the law. The poll found 70 percent of likely voters in the state approve of it.

Maricopa County Sheriff Joe Arpaio, known for his aggressive anti-illegal immigrant programs, told Fox News that the people of Arizona are "fed up" with violence and the "flood of illegal immigrants" in their state. But he, too, predicted the state legislation would prompt Washington to act.

"They want something done about it. Congress and the federal government has done nothing," Arpaio said. "But I predict that Congress will do something now."

Sen. Richard Shelby, R-Ala., said on "Meet the Press" that Arizona passed its law out of frustration that the federal government was not doing enough to enforce immigration law.

He would not say whether he'd support a federal immigration package in the months ahead.

"I think we'd have to look at the details," he said.

Other Republicans were wary.

"It's not a great time to take this issue up in Washington," Senate Republican Leader Mitch McConnell told "Fox News Sunday." "I think it's an important issue. ... But of course, now we have a very high unemployment rate. I just don't think this is the right time to take up this issue."

Sen. Saxby Chambliss, R-Ga., said on CNN's "State of the Union" that negotiations over the federal budget and Wall Street regulation package will "consume an extensive period of time." He said Arizonans were in their right to address the issue at the state level, but said, "I'm not sure where you find the time to deal with these other major issues" in Washington.

But Sen. Robert Menendez, D-N.J., said there's enough time to deal with both immigration and climate change legislation in this session, as well as the other issues. He denied that Democrats were pursuing the former just to drive Hispanic voters to the polls.

"It's beyond Latinos. It's for everybody in this country," he said.

Some Republicans don't buy that argument. Sen. Lindsey Graham, R-S.C., wrote in a scathing letter to Senate leaders Saturday that rushing to complete an immigration package amounts to "nothing more than a cynical political ploy."

"Expecting these major issues to be addressed in three weeks ... is ridiculous. It also demonstrates the raw political calculations at work here," Graham wrote. "Let's be clear, a phony, political effort on immigration today accomplishes nothing but making it exponentially more difficult to address in a serious, comprehensive manner in the future."

Graham's letter was a bombshell for Senate leaders because his support on a months-in-the-making climate change package was critical.

Graham said Saturday he would withdraw from the climate change negotiations if immigration moves to the front of the legislative line -- shortly afterward co-author Sen. John Kerry, D-Mass., announced that he was putting the climate bill on hold.

In the face of the backlash, Senate Majority Leader Harry Reid stood by the call to push forward on immigration legislation.

"I appreciate the work of Senator Graham on both of these issues and understand the tremendous pressure he is under from members of his own party not to work with us on either measure," he said in a statement. "But I will not allow him to play one issue off of another, and neither will the American people. They expect us to do both, and they will not accept the notion that trying to act on one is an excuse for not actingon the other.