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Mr. Obama also made it clear that he contemplated leaving relatively few troops in Afghanistan after the NATO combat mission ends in 2014, saying that the mission will be focused on advising and supporting Afghan troops and targeting the remnants of Al Qaeda.

"That is a very limited mission, and it is not one that would require the same kind of footprint we've had over the last 10 years in Afghanistan," Mr. Obama said, standing next to Mr. Karzai at a joint news conference at the White House.

Mr. Karzai professed to be comfortable with that, saying it was up to the United States to decide the size of a residual force. "Numbers are not going to make a difference in the situation in Afghanistan," he said, noting that it was the nature of the broader relationship that mattered.

Addressing a major sticking point between the countries, Mr. Karzai said the United States had agreed to turn over control of the prisons that house terrorism suspects to Afghan control. That would happen, he said, "soon after" he returned to Kabul.

Mr. Obama said that in order to leave any troops behind, the United States would require guarantees of legal immunity for its soldiers – a demand that the administration failed to obtain from Iraq, leading Mr. Obama to withdraw all remaining American troops from that country in 2011.

Mr. Karzai, citing the agreement to transfer detention centers and the planned withdrawal of American troops from Afghan villages, said he would push for such legal immunity.

"With those issues resolved, as we did today, I can go to the Afghan people and argue for immunity for U.S. troops in Afghanistan in a way that Afghan sovereignty will not be compromised, in a way that Afghan law will not be compromised."

In a joint statement released before the news conference, Mr. Obama and Mr. Karzai extolled the progress made by the Afghan security forces, noting that Afghan troops now take the lead in providing security in 80 percent of the country – a number that will rise to 90 percent by spring, when American and NATO troops are scheduled to move to a purely advisory role.

At that time, the statement said, American soldiers will pull out of patrols in villages, a measure that Mr. Karzai had sought.

Both leaders declined to be drawn into a discussion of the specific number of troops who would be involved in either the coming drawdown or in the residual force that would remain in Afghanistan after 2014. Mr. Obama said he would make decisions in the coming months based on the recommendations of his military commanders.

As the White House examines options for the size of a residual force, ranging from roughly 3,000 to 9,000 troops, Mr. Obama has directed his advisers to answer a basic question: Is such a force necessary to carry out the narrow counterterrorism objective and training mission the United States envisions for postwar Afghanistan?

Mr. Karzai came to Friday's meeting with far different expectations, according to Afghan officials.

Although he has been careful not to discuss specific troop numbers in public, Mr. Karzai appears to be counting on a substantial residual American force — perhaps as many as 15,000 troops, whose mission would be to advise Afghan security forces in their fight against the Taliban insurgency and carry out raids against Al Qaeda.

And he is hoping the United States will supply the Afghan Army with the latest military hardware, including tanks and fighter planes.

These very different expectations, analysts said, could reignite the tensions in a relationship between Mr. Obama and Mr. Karzai that has been notoriously fraught over issues like corruption, civilian casualties and threats to Afghan sovereignty.

"There's been a steady rollback of our objectives of what's good enough in Afghanistan," said Vali Nasr, a former senior State Department official who worked on Afghanistan and Pakistan and is the dean of the School of Advanced International Studies at Johns Hopkins University.

"If you're Karzai, you're basically now facing the same calculation that Maliki did in Iraq," said Mr. Nasr, referring to Nuri Kamal al-Maliki, the Iraqi prime minister. " 'If you're not willing to stay in large numbers, why do I need you?' "

The possibilities for friction are compounded by cynicism on both sides: the sense in Washington that Mr. Karzai is a mercurial, unreliable partner, and the suspicion in Kabul that the Americans care about Afghanistan only when they need it for other purposes, like fighting Al Qaeda.

Gen. Stanley A. McChrystal, the former commander in Afghanistan, warned in a recent interview that if the United States shrank too radically its advisory support for the Afghan military or curtailed its civilian programs, it might lose Afghan support for the counterterrorism operations it might want to continue there.

"If you don't have the support of the Afghan people," he said, "there's no reason for them to be supportive of this."

It is a measure of the potential disconnect between Mr. Karzai and Mr. Obama that Afghan officials involved in preparing his trip said the belief in the Afghan leader's inner circle was that Mr. Karzai was coming to the talks with the upper hand.

In Mr. Karzai's view, these officials and other people close to the president said, the United States needs a robust American presence in Afghanistan after 2014 to keep Al Qaeda off balance and Iran and Pakistan at bay. The Afghans "think they are indispensable; they think they have all the leverage," one Afghan official said.

The Afghans were readying complaints about the United States' continuing to detain Afghans at a prison next to Bagram Air Base, north of Kabul, which was supposed to have been handed over to Afghan authorities in September.

Some of Mr. Karzai's advisers are aware of the tenuous support at the White House for the war effort. Fearful of a repeat of Iraq, where Mr. Obama ordered a total military withdrawal, they have prevailed on the Afghan president to soften his position on granting immunity to any American troops stationed in Afghanistan after 2014, which he has done in recent months.

The ideal outcome for Mr. Karzai, these officials said, would be an American and allied training force that would help the Afghan Army make the most of the billions in aid it is expecting to receive, and a robust counterterrorism force that could work with Afghan Special Forces to combat the remnants of Al Qaeda.

The White House's calculation looks very different. While cost was a consideration in Iraq, Mr. Obama is more sensitive now to the budget consequences of keeping troops in Afghanistan after his bruising fiscal showdown with Congress, and the prospect of huge mandatory cuts in the Pentagon's budget.

"You've got to step back and see the whole field from the point of view of taxpayer spending," said Tommy Vietor, a spokesman for the National Security Council.

To some extent, officials said, the administration's floating of a "zero option" for troops is a bargaining ploy, with both the Pentagon and with Mr. Karzai.

But resolving the differences between the United States and the Afghan government, if they are resolved, is likely to have a profound effect on the long-term commitment of other NATO members.

An allied official said that if the White House opted for a minimal troop presence, the rest of the NATO allies were expected to follow suit, especially since the war was even more unpopular among their publics. Many critics complain that Mr. Karzai has been slow to pursue corruption. But some analysts said the emerging American policy might deepen Mr. Karzai's insecurities.

"We will not be able to give Karzai any more spine to go after cronyism and nepotism," said Vanda Felbab-Brown, an expert on Afghanistan at the Brookings Institution.

The one issue on which both sides appear in sync is the peace process, according to the Afghan officials and people close to the Afghan leader. Like their American counterparts, Afghan officials want to see negotiations with the Taliban make progress and offer rosy public assessments of the diplomatic effort.

But privately, the Afghans are aware that there have been no meaningful engagements with the Taliban in nearly a year, and that until the insurgents prove willing to sit down, the peace process will make little headway.