Iraq’s fragile coalition government failed again Saturday to reach agreement on the formation of a cabinet or on whether to ask some U.S. forces to remain beyond December — leaving the Obama administration with an ever-shorter timetable to complete the withdrawal or manage the political fallout from staying.

President Jalal Talabani, who had called political blocs together for the second time in recent weeks, said he gave all parties another two weeks to discuss the issues before reconvening.

Although the administration has informally suggested some numbers, indicating that leaving 10,000 U.S. troops behind — out of the 46,000 still there — might be reasonable, such estimates remain “guesswork,” a senior U.S. military official said.

“The Iraqis haven’t come to any consensus about what it is they might need” in terms of tasks they want the Americans to perform, said the official, one of several who discussed the sensitive issue on the condition of anonymity. Until they do, he said, the Americans are unable to calculate “how many troops that might require.”

Administration officials said that Prime Minister Nouri al-Maliki has been telling some of his own officials that President Obama wants to leave as many as 30,000 troops in Iraq, a figure the White House dismisses as wildly high.

U.S. troop fatalities: June was the deadliest month in two years for U.S. troops in Iraq. (Alicia Parlapiano/The Washington Post)

“Any post-2011 troop presence is going to be significantly smaller than what we have there now,” an administration official said.

Some U.S. officials have come to the conclusion that the Iraqis may never reach agreement on the subject before the last troops are scheduled to leave by the end of the year.

“This is not an inevitability,” said a second administration official. “And it’s also not inevitable that we would agree to what they are asking. How this is all going to sort out, we just don’t know.”

In the meantime, the indecision complicates an already vexing problem for Obama.

Despite his pledge of complete withdrawal, the administration has made clear its willingness to continue tasks such as training, air defense, intelligence and reconnaissance, as well as joint counterterrorism missions with Iraqi forces at a time of Iranian inroads, increased violence and ongoing political instability. The 15 U.S. troops killed in Iraq last month marked the highest level in two years; two more were killed by a roadside bomb Thursday.

Current U.S. missions could be carried out with a smaller force and some could be carried out from bases in a third country, officials said.

But the longer the decision takes, the less time Obama has to explain to the American public the importance of preserving a presence, and the more he risks clouding an election-year message that he has overseen the end of the Iraq war.

Obama separated himself from the Democratic field in the 2008 primaries largely by his opposition to the war, and he has withdrawn more than 100,000 troops from Iraq since taking office.

“We take comfort in knowing that the tide of war is receding,” Obama told a prime-time audience last month in a speech that described the dwindling U.S. presence in Iraq and his troop withdrawal plans in Afghanistan.

Over the years, Democrats have provided the strongest opposition to the Iraq war. Last week, Senate Majority Leader Harry M. Reid (D-Nev.) told the Associated Press, “There is no question that the United States must continue to provide support for the Iraqis as they progress, but now is the time for our military mission to come to a close.”

Apart from the politics in both countries, the military has grown increasingly impatient.

“Our concern is that the Iraqis have got to make a decision very, very soon,” the senior military official said. “Toward the end of the summer, major muscle movements are going to have to occur to move 46,000 troops and equipment back home.”

Until Obama decides otherwise, military orders are for complete withdrawal by the end of December.

“The point at which we have flexibility is fast diminishing,” the military official said.

The president can change orders at any time, he said, but the later that happens “it lays on a lot more cost and stresses the logistic system.”

The U.S. Embassy in Iraq has its own concerns about the troop decision and its timing. U.S. military forces provide a substantial portion of security for American civilians, and the State Department plans to gradually turn over those responsibilities to hired contractors.

“There is some concern” about the potential for a more abrupt shift from military protection, U.S. Ambassador James F. Jeffrey said in an interview. “And the Iraqis are concerned: ‘Well, what will happen tomorrow?’ ”

Administration frustration with Iraq’s indecision is tied to concern over the Iraqi government itself. Maliki is in his second elected term, taking office both times after a drawn-out period of political negotiations among competing sectarian interests that left his government challenged from within.

Last year, the United States helped negotiate a deal under which Maliki remained as prime minister in a power-sharing arrangement among his Shiite-dominated bloc, the main Sunni-dominated bloc led by former prime minister Ayad Allawi, and the Kurdish coalition.

Disagreements between Maliki and Allawi over how the deal is being implemented, and within the Shiite bloc that includes the party of Shiite cleric Moqtada al-Sadr, have meant that several cabinet seats remain vacant — including the ministers of defense and interior, who would be important players in any U.S. troop decision.

Talabani, the Kurdish president, has tried without success to broker an agreement on the political and military withdrawal issues. Aside from the Sadrists, who are opposed to an ongoing U.S. presence, the political blocs “all want someone to cross the line first, so they can all cross the line,” one Iraqi official said.

“The situation is still under active consideration,” said Samir Sumaidaie, Iraq’s ambassador to Washington. “But there are certain political steps that have to be taken before the current consensus will jell into a decision.”

Staff writer Ed O’Keefe and special correspondent Aziz Alwan contributed to this report from Baghdad.