But many Iraqis and President Obama's anti-war base oppose any extension of the military's presence even though Baghdad will get the final say.
Iraq, which can make a formal request for an extension, is not expected to decide until September at the earliest when the 46,000 U.S. forces left in the country had hoped to start heading home.
"Any request to keep troops in Iraq beyond the agreed upon withdrawal date of December 2011 would have to come from the government of Iraq and would be given serious consideration by this administration," another U.S. official told Fox News.
Baghdad officials will be weighing questions about the readiness of Iraqi security forces against fears of renewed militant attacks and unrest if U.S. soldiers stay beyond the December pullout deadline.
Keeping 10,000 troops in Iraq after the end of the year is the "high watermark," probably the maximum of what the U.S. would be willing to do," the source told Fox News.
The other official added that a troop drawdown does not represent "disengagement."
"Under the terms of the U.S.-Iraq Strategic Framework Agreement, we are building a dynamic partnership with the Iraqi government and people that features cooperation in a range of sectors, including education, energy, trade, health, culture, information technology, law enforcement and judicial cooperation," the official said.
The White House has worked out options to keep between 8,500 and 10,000 active-duty troops to continue training Iraqi security forces during 2012, according to senior Obama administration and U.S. military officials in interviews with The Associated Press. The figures also were noted by foreign diplomats in Baghdad briefed on the issue.
All spoke on condition of anonymity to frankly discuss the sensitive matter during interviews over the past two weeks.
White House spokesman Jay Carney on Tuesday said the Pentagon is still planning for all U.S. troops to withdraw by year's end, noting that time is running out for Iraq's government to ask them to stay.
"We have said for a long time now if the Iraqi government asks us to maintain some level of troops beyond that end of the year deadline, we would consider it," Carney told reporters in Washington.
He appeared to back off that possibility, however, adding: "That doesn't necessarily mean we would do it. We would just consider it. And I really don't have any more information on that possible outcome because, again, we haven't even gotten a request."
Any change in the U.S. military withdrawal timetable in Iraq -- after more than eight years and more than 4,450 U.S. military deaths -- could open up difficult political confrontations for Obama as pressure builds to close out the Iraq mission and stick to pledges to draw down troops in Afghanistan.
The Senate's top Democrat, Sen. Harry Reid, told the AP that the high cost of keeping U.S. troops in Iraq -- given a mounting U.S. debt crisis and Iraq's fledgling security gains -- is no longer necessary.
Reid, the Senate majority leader, estimated nearly $1 trillion has been spent in Iraq since the U.S. invaded in 2003, including $50 billion this year alone.
"As Iraq becomes increasingly capable, it is time for our own troops to return home by the end of the year and for these precious resources to be directed elsewhere," Reid, Democrat of Nevada, said in the statement. "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."
Reid was responding to a request for comment after 15 U.S. soldiers were killed in Iraq in June, mostly by Shiite militias, in the deadliest month for the American military here in two years. It was the first public statement by a top party leader to oppose Obama's policy in Iraq, and may signal splintering Democratic support over his war planning just as he ramps up his 2012 re-election campaign.
Iraq has flown under Washington's political radar for much of the past year, and Democrats who want Obama to end the war this year as promised vowed to exert more pressure on the White House.
"With a false declaration that combat operations are over in Iraq, what is now Operation New Dawn has ironically become a forgotten war," said Ashwin Madia, a former Marine who served in Iraq in 2005-06 and is now interim chairman of VoteVets.org. "That is about to change."
The group has raised millions of dollars for Democratic Party candidates.
Though violence has dramatically dropped from just a few years ago, when Iraq teetered on the brink of civil war, attacks still happen almost daily. On Tuesday, Iraqi police said at least 35 people were killed when two bombs exploded outside a city council headquarters just north of Baghdad.
Running for president in 2008, Obama promised to withdraw all troops from Iraq -- what he had described years earlier as "a dumb war, a rash war." Shortly after he took office, he pledged to stick to a Dec. 31, 2011, deadline negotiated between Washington and Baghdad for all U.S. forces to leave Iraq.
Recently, however, the door gradually has been opening to push the deadline. In May, former Defense Secretary Robert Gates signaled Obama was willing to keep troops in Iraq beyond December. Last week, Navy Vice Adm. William McRaven, nominated to command U.S. special operations forces, said a small commando force should remain.
Without a request from Iraq, fewer than 200 active-duty troops would stay at the U.S. Embassy in Baghdad as military advisers, a role that is common for American diplomatic missions worldwide. More than 166,000 U.S. troops were in Iraq in October 2007, the peak of the Pentagon's surge.
Prime Minister Nouri al-Maliki refuses to publicly endorse a troops' extension. One of his critical political allies -- a Shiite movement headed by anti-American cleric Muqtada al-Sadr -- has threatened widespread violence if troops stay. Al-Sadr's militias once waged fierce attacks on U.S. forces.
Some of Iraq's Sunnis also oppose an extension. The Sunni Islamic Party in Iraq's northern Ninevah province, in a statement this week, called allowing the so-called "occupation forces" to remain "a great mistake against Iraq and its people."
President Jalal Talabani plans a meeting as early as this week of Iraq's political leaders to discuss the troop issue -- which al-Maliki says he does not want to make alone.
The U.S. will not keep thousands of troops in Iraq without immunity. But it's far from certain parliament will approve it. Iraq is still seething from the 2007 shooting by guards from the security firm then called Blackwater Worldwide, which left 17 people dead but could not be prosecuted by Iraq courts because of an immunity deal at the time.
Fox News' Mike Emanuel and The Associated Press contributed to this report.