When President Obama signed the huge economic stimulus bill earlier this year, $1 billion was set aside to help local and state police avoid layoffs or keep their police academy classes intact.

The response was staggering: Departments applied for more than $8.3 billion in aid, meaning only a fraction of the demand can actually be met.

July will be a nervous month for mayors and police commanders as they await official word on how much aid they will get from the grant program known as COPS. The first award announcements are expected this month.

"You've got to cross your fingers and remain optimistic," said Mayor Ron Dellums of Oakland, Calif. He said that without federal aid, his city could lose 140 police positions, and California law gives few options for raising taxes to keep those officers.

Even before a single COPS grant check has been mailed, Dellums said the huge demand for help showed that without more aid, Oakland and other cities "are going to be confronted with the stark reality that we have to cut back."

In Pontiac, Mich., Police Chief Valard Gross has seen plenty of spending cuts in recent years with the downturn of the auto industry. He is worried that the red ink spilling across local budgets everywhere else means his city will get less. "It concerns me greatly," he said. "I can't say what areas are most deserving, but I believe we've been hit harder than just about anyone in the country."

Pontiac's police force has shrunk by about half in the last five years, down to about 70 full-time officers, Gross said. "We're already in the mode where it's an emergency, but we've been able to reorganize, and my guys are still kicking butt." The chief has applied for $4 million to fill 40 positions.

In Philadelphia, the city has applied for $43.2 million in COPS grant funding, which would provide 200 additional police officers. Applications went out in April, and the city is expecting to hear back soon, said Luke Butler, Mayor Nutter's deputy press secretary.

Nutter's five-year plan does not call for any layoffs of police, Butler said, so any new money would fund new positions, not save jobs in danger.

More than 7,200 aid applications poured into the Justice Department and, taken together, nearly 40,000 officers could be laid off without federal help, officials say.

There is no way to verify the number; it depends upon the political process in many places, and law-enforcement officials are not above presenting their potential losses in stark terms that are not necessarily inevitable.

Associate Attorney General Tom Perrelli, the No. 3 official at the Justice Department, said this year's response to the grant program "provided us with a true understanding of the difficulties facing law-enforcement departments today."

Cities are scrambling to keep police on the beat without raising taxes. In St. Louis, officials recently said they might have to cut 105 police positions if the department does not get enough federal aid. If those cuts are made, the city's police force would be smaller than it has been in nearly a century.