SACRAMENTO, Calif. -- It is a wedge issue that pits public safety on the highways against immigration politics. And on Wednesday, round two began on the hotly debated idea of giving California driver's licenses to those living in the state illegally.

Sen. Gil Cedillo, D-Los Angeles, announced a new measure Wednesday that he says should satisfy Gov. Arnold Schwarzenegger's concerns about homeland security by requiring criminal background checks.

"I think the governor should move rapidly to embrace this bill," Cedillo said.

Schwarzenegger spokesman Rob Stutzman said that's not the case.

"Curiously, this week, when we should be focusing on the budget, Sen. Cedillo and the Speaker have chosen to draw a line in the sand, this first week in June," Stutzman said.

Stutzman appeared annoyed that democratic lawmakers have engaged in the same tactics the governor has used -- using deadlines to force action. Last November, based on assurances that Schwarzenegger would work on a replacement bill, the Legislature repealed a driver's license bill signed by former Gov. Gray Davis. But there has been no agreement since.

KCRA 3 reporter Kevin Riggs: "Is the governor setting the bar so high on this issue that he'll never be satisfied? That there'd be no bill that would reach his desk?"

"No, I think quite to the contrary. The governor would still very much like to sign a bill that addresses this issue and solves it," Stutzman said.

But democrats are growing impatient, and so is Los Angeles Police Chief William Bratton, who supports Cedillo's bill.

"We do not have significant concerns about the issue of terrorism, as it relates to this issue. I'm sorry, we just don't. And it's a red herring in many respects," Bratton said.

Conservatives, meanwhile, are pressuring Schwarzenegger to abandon the idea.

"If we do such a thing, provide driver's licenses to those who are here illegally, we will have millions more do the same thing," said Sen. Rico Oller, R-San Andreas.

"Is there the political will to do the driver's license this year? I think there is. I think the people around him are advising him not to do so. I think he's serious about trying to find a solution to the driver's license bill," said Assembly Speaker Fabian Nunez.