BAKU/YEREVAN/PARISBAKU/YEREVAN/PARIS (Reuters) - France, the United States and Russia will step up efforts to end fighting between Azeri and ethnic Armenian forces in the South Caucasus by holding talks in Geneva on Thursday, as fears of a regional war grow.

French Foreign Minister Jean-Yves Le Drian said Russian, French and U.S. representatives would also meet in Moscow on Monday to look at ways to persuade the warring sides to negotiate a ceasefire.

"We want everyone to understand that it's in their interest to immediately stop hostilities without conditions and that we start a negotiation," he told the French parliament's foreign affairs committee.

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Le Drian did not make clear whether any Armenian and Azeri representatives would attend but Azerbaijan said its foreign minister, Jeyhun Bayramov, would visit Geneva on Thursday.

The Armenian foreign ministry said Foreign Minister Zohrab Mnatsakanyan would visit Moscow on Monday but gave no details. It ruled out a meeting with Bayramov.

The warring sides have so far ignored ceasefire calls by Paris, Washington and Moscow, which have mediated for nearly three decades in the conflict over Nagorno-Karabakh, a mountain enclave which under international law belongs to Azerbaijan but is populated and governed by ethnic Armenians.

The Azeri and Armenian leaders have also been at odds over their conditions for halting fighting that began on Sept. 27.

More than 360 people have been killed, including 320 military personnel and 19 civilians in Nagorno-Karabakh, and 28 Azeri civilians. They are the deadliest clashes since a 1991-94 war over Nagorno-Karabakh that killed about 30,000.

Azerbaijan says Azeri cities outside the conflict zone have also been attacked. This has taken fighting closer to territory from which pipelines carry Azeri gas and oil to Europe, and prompted British oil company BP (BP.L) to look at tightening security at its facilities in Azerbaijan.

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"We must be attentive that the war between Armenia and Azerbaijan does not become a regional war," Iranian President Hassan Rouhani said in televised remarks.

Iran, which borders both Armenia and Azerbaijan, has been talking to both the former Soviet republics as concern mounts that Turkey, a close ally of Azerbaijan, and Russia, which has a defence pact with Armenia, could be sucked into the conflict.

CEASEFIRE CONDITIONS

France, the United States and Russia are co-chairs of the Organization for Security and Co-operation in Europe's (OSCE) Minsk Group that mediates over Nagorno-Karabakh.

Turkey has accused the group of neglecting the conflict and said it should not be involved in mediation.

Le Drian hit back at Turkey, reiterating accusations - denied by Ankara - that it is involved militarily and saying this fuelled the "internationalisation" of the conflict.

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Azeri President Ilham Aliyev has said his country will hold talks with Armenia only after the acute phase of military conflict ends, and wants Turkey involved in mediation.

He also wants Armenia to set a timetable for a withdrawal from Nagorno-Karabakh and surrounding Azeri territories.

Armenia’s ceasefire conditions are Turkey "discontinuing its engagement" and "the withdrawal of mercenaries and terrorists or their elimination", Armenian Prime Minister Nikol Pashinyan's press service quoted him as telling Time magazine.

In comments to Sky News, Pashinyan said Turkey and Azerbaijan were pursuing a policy of genocide and "reinstating the Turkish empire". Both have dismissed such accusations in the past.

Some 1.5 million Armenians were killed under Ottoman rule between 1915 and 1923. Turkey accepts that many Armenians living in the empire were killed in clashes with Ottoman forces during World War One, but contests the figures and denies the killings were systematically orchestrated and constitute a genocide.

(Additional reporting by Tuvan Gumrukcu in Ankara, Gabrielle Tétrault-Farber, Dmitry Antonov and Maria Kiselyova in Moscow, Guy Faulconbridge in London and Michele Kambas in Athens, Writing by Margarita Antidze and Timothy Heritage; Editing by Kevin Liffey, Giles Elgood and Gareth Jones)