(Adds further details of conflict, Kerry, quotes)
* Ayatollah Sistani urges followers to fight Sunni insurgents
* ISIL advances into Diyala province near Iran border
* U.S. won't "rule anything out", bar troops on the ground
* ISIL taking arms seized from Iraqi army into Syria -monitors
* Global oil price surges, ISIL surrounds biggest refinery
BAGHDAD, June 13 (Reuters) - Iraq's most senior Shi'ite Muslim cleric urged followers to take up arms against a full-blown Sunni militant insurgency to topple Shi'ite Prime Minister Nuri al-Maliki, escalating a conflict that threatens civil war and a possible break-up of the country.
In a rare intervention at Friday prayers in the holy city of Kerbala, a message from Grand Ayatollah Ali al-Sistani, who is the highest religious authority for Shi'ites in Iraq, said people should unite to fight back against a lightning advance by militants from the radical Islamic State of Iraq and the Levant.
Fighters under the black flag of ISIL are sweeping south towards the capital Baghdad in a campaign to recreate a medieval caliphate carved out of fragmenting Iraq and Syria that has turned into a widespread rebellion against Maliki.
"People who are capable of carrying arms and fighting the terrorists in defense of their country ... should volunteer to join the security forces to achieve this sacred goal," said Sheik Abdulmehdi al-Karbalai, delivering Sistani's message.
Those killed fighting ISIL militants would be martyrs, he said as the faithful chanted in acknowledgement.
U.S. President Barack Obama threatened military strikes against ISIL on Thursday, highlighting the gravity of the group's threat to redraw borders in an oil-rich region which is sending shockwaves through the Middle East.
Amidst the spreading chaos, Iraqi Kurdish forces have seized control of Kirkuk, an oil hub just outside their autonomous enclave that they have long seen as their historical capital, three days after ISIL fighters captured the major city of Mosul.
There are now concerns that sectarian and tribal conflict might dismember Iraq into Shi'ite, Sunni and Kurdish entities.
Reflecting fears that ISIL's insurgency could mushroom into civil war and disrupt oil exports from a major OPEC member state, brent crude edged further above $113 a barrel on Friday, up about $4 since the start of the week.
MALIKI MUST ACT, KERRY SAYS
Obama said military action alone was no panacea against ISIL and alluded to long-standing Western complaints that Maliki has done little to heal sectarian rifts that have left many of Iraq's minority Sunnis, cut out of power since Saddam Hussein's demise, aggrieved and vengeful - a mood exploited by ISIL.
"Prime Minister Maliki and all of Iraqi leaders need to do more to put sectarian differences aside," U.S. Secretary of State John Kerry said at a news conference in London on Friday.
The ISIL advance has been joined by former Baathist officers who were loyal to Saddam as well as disaffected armed groups and tribes who want to oust Maliki.
Cities and towns that have fallen to the militants so far have been Sunni and the gains have largely been uncontested.
It had long been known that Mosul, a city of two million people, harbored not just ISIL but also the Baathist militant group the Naqshbandi Army, believed to be headed by Ezzat Ibrahim al Douri, a former close aide to Saddam.
After the fall of Saddam to the U.S.-led invasion in 2003, officers from the old Iraqi army who had not been reconciled to the new order collected in the Mosul area. The city's proximity to the border with Syria allowed Baathists - Saddam's political party - and Islamic radicals freedom of movement.
The pace of events means that now, an alarmed Shi'ite Islamic Republic of Iran, which in the 1980s fought Saddam for eight years at a time when the Sunni Iraqi leader enjoyed quiet U.S. support, may be willing to cooperate with the "Great Satan" Washington to bolster mutual ally Maliki.
The idea is being discussed internally among the Tehran leadership, a senior Iranian official told Reuters, speaking on condition of anonymity. "We can work with Americans to end the insurgency in the Middle East," the official said, referring to the sudden escalation of conflict in Iraq.
Thrusting further to the southeast after their seizure of Mosul in the far north and Saddam's hometown of Tikrit, ISIL entered two towns in Diyala province bordering Iran.
Saadiyah and Jalawla had fallen to the Sunni Muslim insurgents after government troops fled their positions.
Iraqi army units subsequently subjected Saadiyah and Jalawla to artillery fire from the nearby town of Muqdadiya. ISIL fighters eventually withdrew from Jalawla and well-organized Kurdish Peshmerga fighters took over.
Iraqi army helicopters fired rockets at one of the largest mosques in Tikrit on Friday, according to witnesses. There were no further details available.
"CHANCE TO REPENT"
Obama said on Thursday he was considering "all options" to support Maliki's central government that took full control when the U.S. occupation wound up in 2011.
"I don't rule out anything because we do have a stake in making sure that these jihadists are not getting a permanent foothold in either Iraq or Syria," Obama said at the White House, when asked whether he was contemplating air strikes.
Giving a hint of their vision of a caliphate, ISIL published sharia rules for the realm they have carved out in northern Iraq, including a ban on drugs, alcohol, cigarettes and an edict on women to wear only all-covering, shapeless clothing.
ISIL militants were reported to have executed soldiers and policemen after their seizure of some towns.
On Friday, ISIL said it was giving soldiers and policemen a "chance to repent ... For those asking who we are, we are the soldiers of Islam and have shouldered the responsibility to restore the glory of the Islamic Caliphate".
Residents near the border with Syria, where ISIL has exploited civil war to seize wide tracts of that country's east, watched militants bulldozing tracks through frontier sand berms - as a prelude to trying to revive a medieval entity straddling both modern states.
ISIL has battled rival rebel factions in Syria for months and occasionally taken on President Bashar al-Assad's forces.
ISIL's Syria branch is now bringing in weapons seized in Iraq from retreating government forces, according to Rami Abdulrahman, head of the Syrian Observatory for Human Rights monitoring group.
But its fighters appear to have held back in Syria this week, especially in their eastern stronghold near the Iraqi border, while their Iraqi wing was making rapid military gains.
Matthew Henman, Head of IHS Jane's Terrorism and Insurgency Center said in a report that ISIL's capture of Iraqi territory along the Syrian border will give the group greater freedom of movement of men and material across the two countries.
"Light and heavy weaponry, military vehicles, and money seized by ISIL during the capture of Mosul will be moved into desert area of eastern Syria, which ISIL has been using as a staging ground for attacks," he said.
At Baiji, near Kirkuk, ISIL fighters ringed Iraq's largest refinery, underlining the incipient threat to the oil industry.
Further south, militant forces extended their advance to towns only about an hour's drive from Baghdad, where Shi'ite militia were mobilizing for what could be a replay of the ethnic and sectarian bloodbath of 2006 and 2007.
Trucks carrying Shi'ite volunteers in uniform rumbled towards the front lines to defend Baghdad.
SADR HOLDS FIRE
Despite the call to arms from Sistani, influential Shi'ite cleric Moqtada al-Sadr, who led revolts against U.S. forces, has not called on his followers to mobilize. At Friday prayers, his faithful were instructed to wait for directions in the coming days on how to form "peace regiments" that will be tasked with defending holy sites.
A source in Sadr's media office said he did not approve of mobilization and warned against would-be volunteers who he branded "fitna", who want to drag Iraq into destructive sectarian warfare.
Maliki's army already lost control of much of the Euphrates valley west of the capital to ISIL last year. With the evaporation of the army in the Tigris valley to the north, the government could be left with just Baghdad and areas south - home to the Shi'ite majority in Iraq's 32 million population.
ISIL and its allies took control of Falluja at the start of the year. It lies just 50 km (30 miles) west of Maliki's office.
ISIL has set up military councils to run the towns they captured. "'Our final destination will be Baghdad, the decisive battle will be there' - that's what their leader kept repeating," said a regional tribal figure.
As with the concurrent war in Syria, Iraq's conflict cuts across global alliances. The United States and Western and Gulf Arab allies back the mainly Sunni revolt in Syria against the Iranian-backed President Assad, but have had to watch as ISIL and other Islamists have come to dominate large parts of Syria.
The global oil benchmark price has jumped in reaction to ISIL's rise, although Iraq's main oil export facilities are in the largely Shi'ite areas of the south and are "very, very safe", Oil Minister Abdul Kareem Luaibi said.
The million-strong Iraqi army, trained by the United States at a cost of nearly $25 billion, is floundering amid poor morale and corruption. Its effectiveness is hurt by the perception in Sunni areas that it pursues the hostile interests of Shi'ites.
(Additional reporting by Oliver Holmes in Beirut, Ziad al-Sinjary in Mosul Isabel Coles in Arbil, and Washington bureau; Writing by Peter Millership; Editing by Mark Heinrich)