Exxon Mobil has asked Iraq's prime minister if it can keep running a huge southern oilfield despite disagreements over rival contracts signed with the country's autonomous Kurdistan region, the government said on Monday.

The face-to-face talks between Shi'ite premier Nuri al-Maliki and Exxon's top executive in Baghdad come as the U.S. major offers to sell its stake in the West Qurna-1 oilfield in the south after clashing with Baghdad over its deals with the self-ruled Kurdish enclave in the north.

Iraq has been clear it considers deals oil companies like Exxon sign with the Kurdish enclave illegal. But the meeting may suggest Exxon is testing its room to balance investments with OPEC-member's central government and those with the self-governed Kurdistan region.

"Exxon Mobil asked to meet with the prime minister to know his opinion on the company's contracts in the south and in the northern region and if there was a possibility to keep working on both contracts," Maliki's media adviser Ali al-Moussawi said after the meeting.

"The prime minister's answer was clear to the head of Exxon that they can't keep operating on both deals at the same time and they should observe Iraq's laws."

A statement from the government said only that Exxon Chief Executive Rex Tillerson had "expressed his company's keenness to continue and expand its work in Iraq."

Iraqi officials had said late last year that China National Petroleum Corp, or CNPC, had emerged as the favourite in negotiations to take over Exxon's 60 percent stake in the $50 billion the West Qurna-1 project.

Iraq's Arab-led central government and Kurdistan Regional Government run by ethnic Kurds are caught in a dispute over control of oil revenues, oilfields and territory that is testing Iraq's federal union.

Iraq's government says it alone has the constitutional authority to export crude oil and sign deals, but Kurdistan says the constitution allows it to agree to contracts and ship oil independently of Baghdad.

Attempts to resolve the dispute have failed in part because of disagreements over a long-delayed oil and gas law meant to set a clearer framework for managing the country's vast oil reserves, the world's fourth largest.