BAGHDAD (AP) -- Iraq's Kurds are holding firm to their claim on the country's presidency and the once-dominant Sunni minority is trying to push for checks on the powers of Shiite Prime Minister Nouri al-Maliki as the country faces a looming parliamentary deadline to choose a new political leadership.

Iraq's top political leaders are meeting for the second day in a row for face-to-face talks to find a way out of the country's eight-month political deadlock since March 7 elections.

But producing a deal by Thursday's scheduled parliamentary session is difficult. Standing in the way are issues such as how to apportion key government posts, what role to give the Sunni-backed Iraqiya coalition led by a senior Shiite politician and the deep distrust between political factions.

Iraq has yet to cobble together a new government after the March vote. That inconclusive election set off a bitter fight between al-Maliki, now partnered with anti-American cleric Muqtada al-Sadr in an Iranian-backed coalition and, on the opposite side, the Iraqiya coalition led by former prime minister Ayad Allawi. The Sunnis see Allawi's coalition as their only hope for political power.

With momentum recently shifting in al-Maliki's favor, the question appears to be what role to give to Allawi's Iraqiya coalition. If the coalition is shut out altogether and the Sunnis left feeling disenfranchised again, the risk is a return to sectarian violence.

But members of Iraqiya have balked at joining a government with al-Maliki at the helm and have been pushing for some way to limit his powers.

During a news conference Tuesday, a senior Iraqiya leader, Saleh al-Mutlaq, said the issue of who will be the next prime minister has not yet been settled and will be discussed in the next few days.

Alluding to Iraqiya's concerns over al-Maliki, he said the next prime minister should be someone who would not repeat the country's dictatorial past and would cooperate with everyone in running the country.

A lawmaker backing al-Maliki, Bahaa al-Aaraji, said the main issue keeping Iraqiya from joining in a power-sharing government with al-Maliki is that they want positions with real power in the next government. This is to serve as a check on al-Maliki, who they say is consolidating his power.

Another politician linked to al-Maliki said Tuesday's talks would focus on the three top positions in the government -- the prime minister, president and parliament speaker -- and how to apportion them.

He said officials are trying to persuade Iraqiya to accept the parliament speaker's post, which Allawi has already rejected, or persuade the Kurds to give up the presidency.

The Kurds control three provinces in Iraq's north and are considered a key partner in an any new government. The official spoke on condition of anonymity due to the sensitivity of the talks.

Both officials said the U.S. has been pressuring the Kurds to give up the presidency in favor of Allawi as a way to keep him in the government but to no avail.

Kurdish lawmaker Fouad Massoum declined to comment on whether there is U.S. pressure, but ruled out the possibility of the Kurds giving up the presidency.

"This is out of the question," he said.

While the Kurds are believed to support al-Maliki because he will guarantee them the presidency, they have yet to declare their intention publicly. Massoum said they are waiting for the meetings of Iraq's top political leaders to finish before announcing their official position.

Monday's meeting in the northern city of Irbil was the first time top political leaders met face-to-face since the March vote. The meeting signaled a willingness to move the stalled political process forward amid deepening frustration among Iraqis and growing criticism abroad, but failed to produce any resolution.

The U.S. has been pushing hard for a government that would include all the major blocs, including Iraqiya. U.S. Sen. John McCain, who met with Iraq's political leaders in Baghdad on Tuesday, said the U.S. was "disappointed" it was taking so long for politicians to enforce the will of the people. The Republican senator, who has been one of the most vocal supporters of the Iraq war, pinned his hopes on the second day of the meetings, and emphasized the need for progress "within days, not months."

"We're disappointed that it's taking eight months," he told reporters after meeting al-Maliki.