Sacramento -- Assembly Speaker Fabian Nunez on Wednesday presented a new plan designed to break the state's week-old budget impasse, offering significant concessions to cities and counties that are demanding constitutional protections against state raids on their tax revenues.

"We're very close to having an agreement I could feel comfortable with,'' said San Jose Mayor Ron Gonzales after a conference call with other mayors. "Rather than getting robbed in the future, we've defined the parameters of the loan."

The main sticking point on reaching a final deal on a $77.6 billion budget revolves around what level of funding protection to give cities and counties. Democrats rejected a deal agreed to in May by Gov. Arnold Schwarzenegger and local governments that would allow the state to cut their budgets $2.6 billion over the next two years in exchange for a constitutional guarantee that no money would be taken after that.

Nunez, a Los Angeles Democrat, presented a plan Wednesday that includes strict limits on how much and when the state could borrow money from local governments. He said he was willing to go only so far to protect city and county funding.

"We have made a lot of concessions to the local governments here, and ... they think they are going to get the whole enchilada, and that's just not the way it's going to work," he said. "They've got to give some things up."

There are still areas of disagreement.

Gonzales said sticking points included the definition of what would constitute the fiscal emergency needed to borrow money from local governments. And the mayors want a requirement that a declaration of a fiscal emergency must be approved by a four-fifths vote of the Legislature -- a condition Democrats are balking at because it would give cities and counties more protection than schools receive.

"That's not even in the ballpark," said Senate President Pro Tem John Burton, D-San Francisco. "They would rather protect their local bailiwick than protect schools."

But Senate minority leader Dick Ackerman, R-Irvine, emerged from negotiations Wednesday night saying that local governments should "dig in'' on their insistence that a four-fifths legislative majority be required for borrowing money from local governments.

Republicans have said they want to see the governor's original deal go forward but were likely to agree to any proposal the municipalities accepted.

Under Nunez's proposal, lawmakers would be able to take local government funds twice every 10 years -- and up to $1 billion each time. Nunez said he would be willing to agree that the state could not cut their funds for three years after the $2.6 billion hit determined in the May deal.

Nunez also said he would accept a proposal that wouldn't allow the state to take more money until any loans were paid off.

"That's huge movement for us," Nunez said. He said municipalities would be able to borrow from private lenders with the state's backing in years money was cut.

Cities also want to get interest on any loan, something Nunez said he was willing to consider.

Burton said he would not wait around and negotiate much longer, saying he would put a budget up for a vote as soon as Friday that takes the $1.3 billion without providing any protection for the future. He said if local governments did not like lawmakers' terms, they should stick with their November ballot measure that would provide complete constitutional protection.

Schwarzenegger told local officials he would be against their measure, and Democrats said they had seen polls showing that the measure was in trouble.