* Parliament session brought forward to Sunday
* Islamic State claim to have executed captured general
* Sunni Militants threatening Baghdad
By Isra'a al-Rubei' and Maggie Fick
BAGHDAD, July 8 (Reuters) - Islamist militants claimed responsibility for suicide bombings in Baghdad, and there were signs the deadlock paralysing Iraq's parliament might finally be loosening in the face of the threat from the "Islamic State" that has seized much of the country.
The Sunni Muslim group, which has taken over large areas of Syria and Iraq, posted web photos of two men with scarves covering their faces, posing in front of its black and white flag and machineguns. It identified them as the Baghdad bombers and said they were Lebanese and Libyan.
Five people were killed in the first blast at a cafe in the Washash district on Sunday night. Four police and three civilians were killed the next day at a checkpoint in Kadhimiya, a northern neighbourhood.
Both districts are predominately Shi'ite Muslim, raising fears the capital could return to the days of sectarian bloodletting from 2006-2007.
Baghdad had seen few attacks compared to the violence in other areas hit by the Islamic State's lightning offensive last month.
Hopes that political paralysis in Baghdad may be broken with the formation of a new government to confront the insurgency were dashed on Monday when parliament delayed its next meeting for five weeks - only to reverse its decision 24 hours later.
On Tuesday the acting speaker of the new parliament Mehdi al-Hafidh said parliament would bring forward the session to Sunday, instead of August 12.
"Any delay in this could jeopardize the security of Iraq and its democratic course and increase the suffering of the Iraqi people," the speaker said.
The postponement had been criticised by the lawmakers themselves, who also blamed each other, and by Washington. The U.S. State Department said the "dire situation on the ground" made progress to resolve the impasse all the more urgent.
With no sign that Shi'ite Prime Minister Nuri al-Maliki will abandon his bid for a third term, his Sunni, Shi'ite and Kurdish opponents warn there is a risk that Iraq will fragment along ethnic and sectarian lines.
The reasons for the deadlock over the nomination of the top three posts in government - prime minister, president, and speaker - mirror the broader divisions in the country.
The Islamic State, an offshoot of al Qaeda, until recently called itself the Islamic State in Iraq and the Levant (ISIL). It is spearheading a patchwork of insurgents who hold territory grabbed in a blitzkrieg across the north and west of Iraq and have threatened to move on Baghdad.
The city of 7 million faces threats from militants on three sides.
Islamic State and other Sunni insurgents - ranging from tribal fighters, more moderate Islamists, ex-military officers and members of the banned Baath party - are in the western Baghdad suburbs and cities to the north. Clashes have also erupted to the south.
The Iraqi military, backed by Shi'ite militias and volunteers, has yet to take back any major city but is trying to advance on Tikrit, the late dictator Saddam Hussein's hometown in Salahuddin province.
The army said on Tuesday it had "cleansed" the road from Baghdad to Samarra, 100 km (60 miles) to the north, but attacks are still being launched on security forces on the road.
On Tuesday a suicide bomber killed three policemen and two civilians 15 km south of Samarra, a local security source and a medic said.
The United Nations said last week more than 2,400 Iraqis had been killed in June alone.
Backed by Shi'ite militias and volunteers, the military has been trying since June 28 to advance into Tikrit and defeat opponents including the Naqshbandi Army, a group led by former army officers and Baathists like Ezzat Ibrahim al-Douri, Saddam's feared deputy and lifelong confidante.
Douri is also from the Tikrit area, and is the most senior of Saddam's aides still at large.
Staunchly opposed to what they see as an exclusionary Shi'ite-led government in Baghdad, veterans of the Baath party that ruled Iraq before 2003 found common cause with Islamic State militants when they overran Tikrit and Mosul, the north's largest city.
The prime minister's military spokesman said however on Tuesday "ex-army officers and tribesmen" in Sunni majority areas like Mosul and Tikrit were rallying against the Islamic State.
"(They) are getting ready for a popular uprising against the Islamic state," Lieutenant General Qassim Atta told reporters. "Our security forces are ready to provide tribesmen with the help they need," he said.
The army has intensified its air raids on militant-held positions in northern provinces including Salahuddin and Nineveh, where Mosul is located, in an effort to drive out the insurgents.
The Islamic State posted photos and a statement on social media late on Monday showing the bloodied corpse of an army officer the group identified as Brigadier General Ibrahim Abdullah Hussein, saying it had executed him.
Hussein was one of the officers the government said had fled Tikrit at the beginning of last month's offensive, which saw army and police flee in the wake of the onslaught. There was no way to independently confirm the claim.
Residents of the city say the militants are holding prisoners at Saddam's former palace there.
Thirty "terrorists" were killed in a government air strike on the town of Sharqat north of Tikrit on Tuesday, Atta said.
A doctor at the local hospital told Reuters five civilians were wounded when a helicopter gunship fired on the hospital, where the doctor said some wounded Islamic State fighters were being treated. (Additional reporting by Ahmed Rasheed; editing by Dominic Evans and Andrew Roche)