Updated

As the House and Senate continue to debate overhauling America's immigration system, new differences are emerging between the two chambers over how to handle criminal illegal immigrants.

The controversy is emerging after two police officers were killed, allegedly by illegal immigrant drunk drivers with prior DUI arrests.

"There are thousands of Americans killed intentionally and accidentally by illegal immigrants who have already been arrested and could have been deported from the U.S.," said Kris Kobach, an attorney representing Immigration and Customs Enforcement agents who claim the Obama administration is preventing them from deporting law-breakers.

In Houston, police say 23-year-old Andres Munos was driving drunk on May 20 when he struck and killed 47-year-old Sgt. Dwayne Polk, a Harris County sheriff's deputy. Munos, who is in this country illegally, was previously arrested in 2010 for drunk driving and unlawful carrying of a weapon.

In Phoenix, Jesus Cabrera-Molina admitted he was drunk and high on cocaine the night his SUV struck and killed Phoenix Police Officer Daryl Raetz -- but he denies he was behind the wheel. Witnesses disagree, and Phoenix police have charged Molina with manslaughter.

Under a House bill now being debated, drivers like Munos and Molina would likely get deported immediately by ICE or local police, who would have the authority to enforce federal immigration statutes.

Under the Senate bill, however, illegal immigrants accused of non-violent crimes are entitled to a hearing and a taxpayer-funded lawyer. Those with three or fewer misdemeanors and some felonies would be allowed to remain if they had children or wives in the U.S.

Advocates say all illegal immigrants must have the opportunity to fairly present their cases in court. The temptation to make immigration laws "tougher" is a mistake, they say -- which is why the Senate bill provides for judicial "discretion" if deporting a criminal alien imposes a "hardship" on a family members living in the U.S.

"These are real human situations that require human decisions," said Crystal Williams of the American Immigration Lawyers Association. "Drunk driving needs to be treated as the crime that it is -- an alcohol-related, a substance abuse-related crime and not something related to immigration. The two are wholly unrelated."

Differences in the House and Senate bills will likely have to be worked out in committee should the respective immigration efforts pass each chamber.