Washington (CNN) -- Latino members of Congress Tuesday called on Arizona's governor to kill a state bill that would require police to determine whether people are in the United States legally, arguing that it unconstitutionally authorizes discrimination.

"When you institutionalize a law like this one, you are targeting and discriminating at a wholesale level against a group of people," Rep. Raul Grijalva, D-Arizona, told reporters. Grijalva and Rep. Luis Gutierrez, the chairman of the Congressional Hispanic Caucus Task Force on Immigration Reform, said Republican Gov. Jan Brewer should veto the measure, which the state Senate approved Monday.

The bill would require immigrants to carry their alien registration documents at all times and require police to question people if there's reason to suspect they're in the United States illegally. It also targets those who hire illegal immigrant day laborers or knowingly transport them.

Brewer, a Republican, has yet to take a public stance on the bill.

Grijalva, whose congressional district runs from the Mexican border to the outskirts of Phoenix, said the legislation "is not just mean-spirited, it is directed at a specific population." Gutierrez, D-Illinois, said the measure infringes on the federal government's authority over immigration laws as well as being discriminatory.

"I'm Puerto Rican. I was born in Chicago, and my family has been U.S. citizens for generations," Gutierrez said. "But look at my face, listen to my voice. I'd probably get picked up in Arizona and questioned. Is that what we want in America?"

Currently, officers can check someone's immigration status if the person is suspected in another crime. Critics argue the new law would foster racial profiling, saying that most police officers don't have enough training to look past race while investigating a person's legal status.

The bill is considered to be among the toughest immigration measures in the nation. Supporters say the measure is needed to fill a void left by the federal government's failure to enforce its immigration laws. Its leading sponsor, state Sen. Russell Pearce, said, "Illegal is not a race, it's a crime."

"We're going to take the handcuffs off of law enforcement. We're going to put them on the bad guy," Pearce, a Republican, told CNN.

"You know, this is amazing to me. We trust officers, we put guns on them, they make life and death decisions every day," he added. "They investigate capital crimes, they investigate sophisticated crimes, but we're afraid they're going to pick up the phone and call ICE [U.S. Immigration and Customs Enforcement]."

The tough rhetoric has angered immigration advocates, such as Isabel Garcia, an Arizona legal defender, who says the legislation "legalizes racial profiling."

"I think this bill represents the most dangerous precedent in this country, violating all of our due process rights," she told CNN. "We have not seen this kind of legislation since the Jim Crow laws. And targeting our communities, it is the single ... largest attack on our communities."

The state Senate's Democratic leadership slammed Monday's vote, saying the bill doesn't truly address Arizona's real immigration problems.

Senate Bill 1070 "is exactly why the federal government must act on immigration reform," said Democratic leader Jorge Luis Garcia in a written statement. "We cannot have states creating a jigsaw puzzle of immigration laws. This bill opens the doors to racial profiling with the provision that allows an officer to ask for citizenship papers from someone who only looks illegal."

The bill is an unfunded mandate that is "turning police officers into ICE agents and opening departments to lawsuits allowed by this bill," said Sen. Rebecca Rios.

Rios was referring to a provision in the bill that allows residents to sue their local governments if they feel the law isn't being enforced effectively.

Gutierrez is a leading supporter of a proposed overhaul of U.S. immigration laws, and said the Arizona issue shows why an overhaul is necessary. He urged President Obama to "put his back into the push" and to let Arizona know that federal law trumps state legislation on the issue.