* Exxon CEO met Iraq PM Monday, Kurd president Tuesday
* Iraq official says Exxon moving on right course
* No details of possible improved Baghad terms
By Peg Mackey and Isabel Coles
LONDON/ARBIL, Iraq, Jan 23 (Reuters) - Exxon Mobil may be moving closer to Baghdad's side in its bitter feud with autonomous Kurdistan, industry sources said, with a sweeter deal to keep it operating in southern Iraq on the table.
Exxon Chief Executive Rex Tillerson has met with Iraqi Prime Minister Nuri al-Maliki to ask about staying at the West Qurna 1 field - a greater potential prize than its deals with the northern Kurds if Baghdad could mould the deal closer to the Kurdish model.
Tillerson met Kurdistan President Masoud Barzani on Tuesday in Davos.
"After the meeting with Maliki, we can now say that Exxon is moving along the right course the central government has set for foreign oil companies working in the country," an Iraqi oil source said on Wednesday, without giving any details.
Last year, Exxon offered to sell its stake in the $50 billion southern oilfield after fighting with Baghdad over its contracts with Kurdistan, deals which the central government in Baghdad rejects as illegal.
"Baghdad wants to lure Exxon in. Will they bite?" said one oil industry source.
Another Western oil executive from a company involved in the south said Exxon would likely need to see improved conditions on its West Qurna deal to stay.
Any shift in Exxon's positioning would be a major win in favour of Baghdad over Kurdistan in their political battle over control of the country's oil resources that is straining Iraq's uneasy federal unity.
Iraqi officials were tight-lipped on details of any offer, but Baghdad has been clear in the past foreign oil companies operating in the south should give up their contracts with the Kurdish region or risk losing assets in the rest of Iraq.
"At the same time we understand Exxon has to fix its issues with the Kurdistan Regional Government before we can reach a final agreement on continuing working in the south," the Iraqi oil official said.
A statement on the website of the Kurdistan Regional Government (KRG) said Tillerson met with Kurdish leader Barzani to discuss the company's operations, without giving any further details.
"You don't engage at CEO level unless there is something substantial on the table," a chief executive of a rival oil company operating in the south said about Exxon's meetings.
Exxon was the first major oil company to test the waters by signing for six blocs with the Kurdistan Regional Government (KRG) in 2011, upsetting Baghdad which says only the central government has the right to oilfield deals and to control exports of the OPEC member's crude.
Its move heralded an increase in tensions between the Arab-led central government and the self-governed region run by ethnic Kurds since 1991, where they have their own armed forces but rely on Baghdad for a share of the national budget.
The two regions late last year sent troops from their respective armies to reinforce positions along their disputed internal border in a major escalation of tensions. Talks to defuse the crisis are ongoing.
It was not clear whether Baghdad's offer was conditional on Exxon putting its Kurdish assets on hold, and further details were not immediately available, but Iraq's oil minister hinted a week ago that sweeter terms could be on offer.
"We wish Exxon would stay and work with us. When we talk about West Qurna, we are talking about double of what is going to be achieved in the Kurdistan region," Abdul Kareem Luaibi told Reuters in an interview.
"We have started our preparations for the fifth licensing round, which is going to be with completely different contract shapes that are going to be very attractive to the oil companies."
Oil majors have been lured north by better contract terms, security and an easier working environment compared with the bureaucracy and infrastructure bottlenecks which hamper oil projects in the rest of Iraq.
Iraqi officials 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 West Qurna-1 project.
Iraq's central government and the KRG have long feuded over control of oil revenues, oilfields and territory, but the Exxon talks come with the country caught in a crisis over power-sharing among Shi'ite Muslims, Sunnis and ethnic Kurds.
Any Exxon offer from Baghdad would emerge at a time when Shi'ite premier Maliki is struggling to manage increasing political pressure from Sunni street protests that have complicated his government's fight with Kurdistan.
While Iraq's government says it alone has the legal authority to export crude and sign deals, Kurdistan says its right to grant contracts and ship oil is enshrined in the constitution.
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 world's fourth largest oil reserves.