The city is about to cut staffing on city fire engines in a move that is putting smoke-eaters and New Yorkers “in greater danger,” union officials tell us.

Engine companies usually have four smoke eaters, but 20 companies citywide currently have a “fifth man” who can act as a wildcard when things get heated and someone needs an extra hand.

“These 20 engine companies are a manpower pool,” said Uniformed Firefighters Association president Gerard Fitzgerald.

But they’re expected to be axed tomorrow, thanks to a byzantine labor contract that grants the companies a fifth man only if the FDNY can keep its medical leave numbers low.

“Because we’ve gone over that agreed upon number they’re gonna reduce those 5-man companies back down to four man companies,” Fitzgerald said. Instead, when the FDNY needs help it will pull a man from a truck company instead, potentially putting New Yorkers and our first responders at greater risk.

“The truck companies will be operating with less than their full assignment — that to me is very dangerous,” he said. “If we’re missing one key component, it could be detrimental.”

The news comes at a particularly busy time for the FDNY. “We’ve broken our run total the last five years straight,” Fitzgerald said.

But the FDNY counters that the union agreed to the contract — and shouldn’t whine now that it failed to hold up its end of the deal.

“The union negotiated this,” spokesman Frank Gribbin said. “For decades we have operated safely and effectively with four firefighters.”

Increasing medical leave costs the city big time, because when a firefighter takes off, someone else has to be paid — usually overtime — to fill the spot, Gribbin said.

He argued that while radio runs are up, many are for medical emergencies and other low-risk calls.

“Last year was a record low in serious fires — the lowest in 30 years,” he said.

The changes were expected to be announced Monday.