More than a year after a judge decided that City Hall owes the pension fund for New Orleans firefighters $17.5 million, city officials may finally be running out of time to hand over a budget-busting check or face being held in contempt of court.

City Council President Stacy Head took the stand in Civil District Court on Tuesday and offered a compromise: She would not commit the council to appropriating the full judgment, but she would support taking $2 million from a small surplus at the Police Department and raising another $1.4 million or so by imposing furloughs at the Fire Department.

While that would cover only a fraction of the $17.5 million, Head said it would be more than enough to keep the pension fund from continuing to shrink.

Civil District Court Judge Robin Giarrusso promptly rejected the offer.

She said the city must “make a lot more movement toward the full $17.5 million,” and she threatened to hold certain officials — she did not name them — in contempt of court. Presumably that could mean fines or even jail for some of the named defendants in the lawsuit, including the mayor, members of his staff and the entire City Council.

Even Giarrusso, however, did not sound optimistic that the council or Mayor Mitch Landrieu’s administration, the two arms of city government responsible for budget decisions, would bend, even with more time to find the extra money.

“I’m giving you some rope that you may well hang yourself with,” Giarrusso said. “I don’t know how else to say that.”

Head and the Mayor’s Office have said repeatedly that taxpayers should not be held responsible for bad investment decisions made by the fund’s independent board of trustees, and that paying the $17.5 million would blow a huge hole in the city’s annual general-fund budget of about $500 million.

The pension fund took a $40.2 million loss on investments last year, according to the latest audit, mostly because of the declining value of real estate and other assets on its books. The value of one property, a golf course in Algiers, tumbled from $39.2 million to $21.7 million in the space of a year.

Still, Giarrusso — backed up by the state 4th Circuit Court of Appeal — has ruled that city officials have no choice: State law says the city must pay whatever the fund’s actuary estimates is needed each year to pay out benefits and invest for future retirees. The city acknowledges having paid far less than that since 2010, when Landrieu took office.

In fact, there are even bigger stakes involved than the $17.5 million, which would cover only the shortfall for 2012. The city’s total bill could climb to $75 million by the end of this year and by tens of millions more next year.

Lawyers for the pension fund have already asked Giarrusso to make the city turn over another $57.6 million to cover what they say is owed for 2010, 2011, 2013 and 2014. They also want the judge to rule that City Hall must appropriate the full $36.2 million the fund’s actuary says is needed for 2015. Giarrusso will hold a hearing on those additional requests in early September.

Whoever is to blame, all sides agree that the pension fund is in terrible shape and is likely to deteriorate further as the standoff with city officials continues.

With its investments in decline, the fund has seen its net assets available for paying benefits shrink from more than $160 million at the end of 2010 to less than $85 million at the end of last year. Landrieu’s administration has refused to buttress the fund with as much money as requested. So to keep paying retirees, the fund’s managers have begun to cannibalize what is left, eating into what’s available to make investments for those still working.

Head offered her compromise as a means of at least arresting that downward spiral during the remainder of the year, presuming the council would act on her proposal as quickly as possible.

She said the fund is bleeding about $691,000 a month, paying out more to retirees than it takes in from working firefighters, the city or other sources. Taking half of a $4 million surplus at the NOPD — available because the city can’t hire as many new officers as budgeted — and imposing furloughs at the Fire Department would more than make up the difference, Head said.

Even as she proposed the stopgap measure, Head argued that more drastic steps are needed to ease the pension crisis. She said the firefighters’ fund should be folded into the pension fund for other municipal employees, which is in better shape, and that firefighters probably need to take less generous benefits.

“We need to come up with a long-term plan,” she said, “because all we are doing right now is propping up, at best, a system that will not survive.”

In any case, Giarrusso dismissed Head’s offer as too small a payment for what’s owed, as well as an unfair burden on the firefighters who would have to take unpaid time off.

“It’s not a big enough step, because the judgment is not for $2 million, the judgment is not for $691,000,” she said, adding, “I don’t care what anyone says, I don’t think you punish the present firemen for the decisions of other firemen.”

The judge’s order was a victory for the head of the local firefighters union, Nick Felton, who has been fighting the city in court for many years over pension benefits and pay issues.

Speaking with reporters after the hearing, Felton would not say exactly where he thinks the city should come up with the extra money.

“I’m not in the business of appropriating,” he said, but the fund’s attorney, Louis Robein, suggested the city could sell bonds to pay off the judgment.