SACRAMENTO -- With little apparent substance to support his optimism, Gov. Gray Davis said Tuesday that he believes progress has been made on the state's budget crisis and expressed hope an agreement could be reached by the end of next week.

Davis' comments came during a ceremony in which he extended a hiring freeze first imposed in October 2001. He also called for eliminating 2,000 unfilled positions in state government.

Combined, the two orders will save the state close to $500 million next year, Davis said.

Davis' confidence on the budget stands in stark contrast to the dark mood in the Capitol one day after the Legislature failed again to meet its July 1 budget deadline.

The state is facing a deficit that could swell to $38.2 billion by next June 30 and without a new budget, state Controller Steve Westly said he cannot pay a number of bills as well as some paychecks. The administration also announced Tuesday that they were sending out an additional 20,000 letters warning workers throughout state government of possible layoffs.

Still, Davis said he has a "growing sense" that Republicans are warming to the idea of using a tax increase to finance the state's existing $10.7 billion deficit.

Legislators are concerned about selling bonds to pay off the existing deficit, Davis said, but added that legislators are coming together because of a "growing recognition that no one will buy those bonds without some dedicated revenue source."

The governor said that while others predict "doom and gloom" and the likelihood of a protracted budget stalemate, he continues "to believe we could see some action by the end of this week or the end of the following week" based on conversations he's had with several party leaders during the past four days.

But a Republican budget negotiator said he knew of no change in the GOP's position on raising taxes.

"We're not going to support a new revenue source or additional sales tax," said Sen. Dick Ackerman, R-Tustin. "There's been no change in our position."

For the third consecutive year, California began the new fiscal year without adopting a new budget. Without an approved plan in place, the state is unable to pay many of its bills including money owed to schools, community colleges, courts, state supplies and others.

As of Tuesday, a long list of non-civil service workers will also go without paychecks -- including elected officials, state appointees and legislative staffers.

While the Legislature is routinely late in adopting a budget, there are signs this year might be different.

California is one of the few states that require the budget be approved by a two-thirds majority of lawmakers. While Democrats hold big majorities in both the Senate and the Assembly, Republicans have more than enough seats to block passage of the budget as long as they want.

With Davis facing a growing recall campaign, some say Republicans have little to gain by settling.

But state finances have never been in such bad shape.

Westly said California has only enough cash to last through the middle of August, and, saddled with the nation's lowest credit rating, there is some question whether Wall Street would loan California any more money without an adopted budget.

In yet another complication of the budget scene, one of the state's largest taxpayer group filed suit Tuesday in Sacramento County Superior Court challenging a hike in the car tax triggered earlier this month by the Davis administration.

Jon Coupal, president of the Howard Jarvis Taxpayers Association, called the $4 billion tax hike -- which will cost the average car owner about $158 annually -- illegal because it was not approved by the Legislature.

Coupal noted that the suit will probably not come before a judge for weeks and a trial might take months to play out.