The Baghdad government, struggling to make ends meet, is unlikely to relinquish such a vital economic prize.
So the demand by Massoud Barzani, who fought a guerrilla war against Saddam Hussein for decades, has raised the specter of a conflict between the independence-minded Kurdish minority and the federal government that could splinter Iraq.
"We will not accept any other solution," Barzani declared Wednesday in the Kurdish city of Irbil. "We want it to be annexed to our region because the majority of its population are Kurds."
The issue of who controls Kirkuk, capital of a province with the same name, has already sabotaged efforts to push an important electoral law through a fractured Parliament, and that could derail a general election scheduled for Jan. 16.
The Kirkuk fields produce one-third of Iraq's oil. If the Kurds get their hands on those reserves, it would almost certainly push some factions in the Shiite majority to take control of the southern fields, which produce the rest of Iraq's oil output.
That would effectively cut out the minority Sunnis, who were the pillar of Saddam's brutal regime but have few resources in their areas in central Iraq, bolstering jihadist insurgents and antagonizing the region's Sunni regimes, particularly in Saudi Arabia and Jordan.
The result would likely be the disintegration of Iraq into sectarian statelets, with the Kurds and Shiites in full control of the country's energy riches.
Iran, which borders the Shiite-dominated south, would probably have access to the southern oil reserves, greatly boosting its own reserves.
At the same time, Turkey and Iran don't want to see an independent Kurdish state emerge in northeastern Iraq. They have enough problems with their own restive Kurdish minorities, and an independent homeland in Iraq would only fuel those groups' separatist ambitions.
If Iraq is to be plunged into another spasm of sectarian savagery, it is likely to explode in Kirkuk, where the long-festering rivalries between Kurd and Arab are already colliding.
Jalal Talabani, a Kurdish leader and veteran guerrilla fighter who is currently Iraq's president, calls Kirkuk "the Kurdish Jerusalem," underlining his people's emotional ties to the ancient city.
The Kurds have long claimed Kirkuk as theirs, saying it was part of the Ottoman-era governate in Kurdistan that existed until World War I. The Arabs, along with the sizeable Turkmen minority in Kirkuk, dispute that.
During Saddam's rule, he forcibly moved rebellious Kurds out of the Kirkuk area as part of his Arabization program. Since Saddam's downfall in the U.S.-led invasion of 2003, the Kurds have been moving large numbers of their people back into the city and its environs in a demographic confrontation.
They want a referendum to decide whether Kirkuk should join three other provinces that make up the Kurdistan Regional Government. So they oppose holding parliamentary elections until a proper census has been conducted.
The prospect of a new sectarian conflict, on top of the brutal bombing campaign being conducted these days by al-Qaida as U.S. forces withdraw and simmering Shiite-Sunni violence, will also impede Baghdad's efforts to attract international oil companies to Iraq to restore the country's rundown oil industry.
Iraq has always been particularly vulnerable to eruptions of violence during times of political deadlock. So as the issue of who is to run the Kirkuk oil fields moves to center stage, the prospect of a new bloodletting is looming closer.
Notable deaths of 2014 [PHOTOS]