Schwarzenegger told reporters about the threats against Mayor Antonio Villaraigosa and Lt. Gov. Cruz Bustamante during a news conference in his office Monday.
Other elected officials of Mexican heritage have also received threats, Schwarzenegger said, but he did not name them.
Bustamante spokesman Steve Green said the lieutenant governor appeared at some immigration rallies with Villaraigosa in March and received "nasty e-mails" afterward. The death threat - "The only good Mexican is a dead Mexican" - came about three weeks ago on a postcard, he said.
Congress is returning from a two-week break and still faces long odds in passing any immigration bill at all, reports CBS News correspondent Bob Fuss. The House passed a bill that sparked protests around the nation; it would make all illegal immigrants into felons and also seek to jail anyone who helps them.
The Senate came close to passing a bill the president likes to let most of the 12 million illegal immigrants become legal. But with opposition from many Republicans and political maneuvering on both sides, efforts to pass it collapsed and it won't be easy to get back on track.
Bustamante and Villaraigosa have opposed the criminalization of illegal immigrants and have said they support legal pathways to helping undocumented workers become citizens.
Schwarzenegger also said he is disturbed that vandals had spray-painted ethnic insults on and torched a Mexican-owned restaurant in San Diego County this month. Officials ruled the April 10 attack a hate crime.
"That is not what California stands for," Schwarzenegger said. "I've asked the district attorneys throughout our state to be vigilant and swift in their actions against those who practice hate against our fellow citizens."
The governor has said he opposes amnesty for illegal immigrants, but he also called a massive deportation of the estimated 11 million foreigners living illegally in the U.S. unrealistic.
Mr. Bush, rebutting lawmakers advocating a law-and-order approach to immigration, said Monday that those who are calling for massive deportation of the estimated 11 million foreigners living illegally in the United States are not being realistic.
"Massive deportation of the people here is not going to work," Mr. Bush said as a Congress divided over immigration returned from a two-week recess. "It's just not going to work."
Mr. Bush also met Tuesday with a bipartisan group of senators at the White House to press his case.
Mr. Bush spoke in support of a stalled Senate bill that includes provisions that would allow for eventual citizenship to some of the illegal immigrants already here. Some conservatives say that would amount to amnesty.
"This is one of the really important questions Congress is going to have to deal with," Mr. Bush said. The president said he thought the Senate "had an interesting approach by saying that if you'd been here for five years or less, you're treated one way, and five years or more, you're treated another."
Standing in the center of a theater in the round-type setting with an audience full of business people, Mr. Bush spoke sympathetically about the plight of foreigners who risk their lives to sneak into the United States to earn a decent wage. He said the U.S. needs a temporary guest worker program to stop people from paying to be smuggled in the back of a truck.
"I know this is an emotional debate," Mr. Bush told the Orange County Business Council. "But one thing we can't lose sight of is that we are talking about human beings, decent human beings."
Mr. Bush said it's important to enforce border laws that are on the books and boasted that 6 million immigrants have been captured and turned back since he took office.
"You can be a nation of law and be a compassionate nation at the same time," he said to applause.