* Parliament's second session postponed amid deadlock
* General killed battling militants 16 km west of Baghdad
* Maliki rejecting pressure to give up premiership (Adds U.S. official, uncertainty about date of next parliamentary session)
By Isra'a al-Rubei'i and Ahmed Rasheed
BAGHDAD, July 7 (Reuters) - Iraq's new parliament put off its next session for five weeks on Monday, extending the country's political paralysis amid a Sunni Islamist insurgency that claimed the life of an army general near Baghdad.
Citing the politicians' failure to reach "understanding and agreement" on nominations for the top three posts in government, the office of acting speaker Mehdi al-Hafidh said parliament would not meet again until Aug. 12.
Putting off the work of reaching consensus is a slap in the face to efforts by Iraq's Shi'ite clergy, the United States, the United Nations and Iran, who have all urged the swift formation of an inclusive government to hold the country together.
"We're looking at a dire situation on the ground, which is why it's so important that things move forward urgently on the ground," State Department spokeswoman Jen Psaki told reporters in Washington in reaction to the delay.
With no signs 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.
"Things are moving faster than the politicians can make decisions," a senior Shi'ite member of parliament told Reuters.
The Islamic State, an al Qaeda offshoot, and a patchwork of Sunni insurgents are holding territory they seized in northern and western Iraq, the majority of it taken last month.
Kurds, who run their own autonomous region in northern Iraq, have taken advantage of the chaos to expand their territory.
Maliki's opponents blame him for last month's defeats and want him to step aside. They accuse him of favouring the Shi'ite majority over the Sunni and Kurdish minorities.
Some hopes remained that parliament would meet again this month, based on local press comments by Hafidh and other comments by a senior U.S. official.
Brett McGurk, deputy assistant secretary of state for Iraq, said on his Twitter feed that Hafidh had clarified that the new date for parliament to resume work was July 13. But parliament would still be required to issue a formal statement to change the date.
Maliki said last week that he hoped to overcome the challenges blocking the formation of a new government after the new parliament's first session ended without agreement on the top posts of prime minister, president and parliament speaker.
The Iraqi military, backed by Shi'ite militias and volunteers, has yet to take back any major cities but is trying to advance on Tikrit, the late dictator Saddam Hussein's hometown in Salahuddin province.
The fighting is taking a heavy toll. The United Nations said last week more than 2,400 Iraqis had been killed in June alone, making the month by far the deadliest since the height of sectarian warfare during the U.S. "surge" offensive in 2007.
A senior Iraqi general was killed in fighting with insurgents near Baghdad on Monday, as the army fights to hold militants back from the capital.
Major General Negm Abdullah Ali, commander of the army's sixth division responsible for defending part of Baghdad, was killed just 16 km (10 miles) northwest of the capital.
A few hours later, four policemen and three civilians were killed by a suicide bomber at a checkpoint in the mainly Shi'ite Kadhimiya district of northern Baghdad.
A bomb exploding at a roadside outdoor cafe killed four people late on Monday in the Nahrawan area just south of the capital, a police officer and a medic said.
Top U.S. defence officials said last week the security forces could defend the capital but would have difficulty going on the offensive to recapture lost territory, mainly because of logistic weaknesses.
In the northeastern province of Diyala, Islamic State militants killed four civilians in the town of Udaim, a police officer said.
Militants killed six civilians including a woman and an elderly man late on Monday in the village of Zawiya near the northern city of Baiji when they tried to arrest a local police officer, an eyewitness said.
According to the witness, Islamic State fighters accused the policeman, who escaped, of trying to form a "Sahwa" or "Awakening" force to rise up against the group.
Sunnis and Kurds blame the National Alliance, the Shi'ite grouping that includes Maliki's State of Law list, for failing to name a prime minister.
Most Sunnis and Kurds walked out of the last parliament, saying they believed the prime minister and president should be chosen along with the speaker as a package, not one at a time.
With parliament's session a day away, they could not resolve the impasse, so the acting speaker postponed the meeting.
Shi'ite MP Haidar Abadi of Maliki's State of Law coalition said it viewed the month-long break as a mistake and said that there would be "strong pressure" for a meeting before Aug. 12, the date set for the next session.
Results of April's elections initially suggested parliament would easily confirm Maliki in power for another term. The loss of the Sunni regions in the north has created an opening for Maliki's opponents, however.
Some within his own alliance are whispering about the need for him to step aside, although Maliki has stated publicly he will not give up his candidacy. (Additional reporting by Ned Parker and Maggie Fick in Baghdad and David Alexander and Missy Ryan in Washington; Writing by Maggie Fick; Editing by Dominic Evans, Tom Heneghan and Lisa Shumaker)