The U.S. oil giant's plan, and the growing defiance of the semiautonomous Kurdistan Regional Government in the northern city of Erbil, is a direct challenge to the authority of Maliki's troubled coalition.
It could also seriously upset Baghdad's ambitious plans to rival Saudi Arabia as the world's top producer over the next decade.
For now, all the signs are that the KRG and the central government, which haven't seen eye to eye for years, are on a collision course.
The gap is widening on a broader front as the Kurds increasingly side with rebel forces fighting in Syria and Baghdad supports the Iranian-backed regime of President Bashar Assad.
"If Exxon Mobil starts drilling operations, Baghdad will have no option but to try and stop them," the Middle East Economic Digest quoted a source in Erbil as saying.
"But they'll have the KRG and the Peshmerga behind them."
The Peshmerga, which means "those who face death," are the Kurds' battle-seasoned fighters who for decades fought a separatist war against Baghdad until Saddam Hussein was toppled in the 2003 U.S. invasion.
Exxon's feud with Baghdad flared in October 2011, when the U.S. company signed an agreement with the KRG to explore six blocks in Kurdistan, including three that lay in disputed territory along Kurdistan's southern boundary.
Kurdistan sits on at least 45 billion barrels of crude, as well as substantial natural gas reserves.
Baghdad declared the Exxon deal illegal, saying the KRG had no authority to make independent deals, and threatened to tear up Exxon's 2009 production-sharing agreement for the huge West Qurna 1 field in southern Iraq.
Exxon, fed up with Baghdad's stingy contracts and endless bureaucratic obstructions, stuck to its guns even though its actions have intensified the rift between Erbil and Baghdad and encouraged Kurdish expectations of independence.
That threatens to splinter the federal state established after Saddam was toppled and to undermine the government's drive to push oil production, recently raised to around 3 million barrels per day, to 10 million bpd later in the decade. National reconstruction depends on achieving that goal.
Maliki, who's facing growing unrest because of his drive to establish one-man rule since U.S. forces departed, even complained to U.S. President Barack Obama.
Washington's worried Shiite Iran is establishing control of its long-time enemy, with its Shiite majority, and could intervene against the Kurds -- whose independence aims it bitterly opposes -- and Exxon and other international companies like Chevron, Total of France and Gazprom Neft of Russia that have also signed exploration deals with Erbil.
Exxon Chief Executive Officer Rex Tillerson flew to Baghdad to meet Maliki in late January but apparently refused to quit Kurdistan.
MEED reports Exxon "has now submitted several staff employment applications to the KRG, despite warnings from Baghdad not to pursue any upstream development.
"The KRG has allocated land for a base camp at Kani Kuchal in the disputed Qara Hanjer exploration and production block, which is near Chemchemaal gas field under development by the United Arab Emirates' Crescent Petroleum and Dana Gas."
Maliki's woes have been multiplied by Turkey, Iraq's northern neighbor. It's stepped in to aid the Kurds by offering to build oil and gas pipelines from Kurdistan to export terminals on the Mediterranean.
The landlocked Kurds are already trucking oil exports northward, bypassing Baghdad's state pipeline network to Turkey.
Although Turkey's dead set against giving its own restive Kurdish minority any measure of self-rule after a 30-year insurgency, the KRG's deep distrust of Baghdad has produced a tactical alliance between energy-hungry Ankara and Erbil.
Baghdad has threatened to take legal action against oil companies exporting crude directly from Kurdistan.
Maliki's even sought to strike a deal with BP, which also has a big stake in southern Iraq, to upgrade the declining oil fields in Kirkuk, a region the Kurds claim.
That's incensed the Kurds, since the operation to boost production from the current 260,000 bpd to 580,000 bpd, would involve fields in contested areas.
Both sides have heavily armed forces confronting each other along Kurdistan's southern border.
Oil could well be the spark to ignite a war.