US Defence Secretary Robert Gates is in the Afghan capital, Kabul, to discuss the future of US involvement there.
He is due to meet Afghan President Hamid Karzai, US troops and allied commanders during his two-day visit.
Later this month, Mr Karzai will announce the schedule for the handover of security responsibility from foreign forces to Afghans.
The visit comes at a time of widespread anger towards the United States over civilian casualties.
On Sunday, President Karzai told Gen David Petraeus, the US commander of foreign troops in Afghanistan, that his apology for the deaths of nine children in a Nato air strike last week was "not enough".
The president added civilian deaths were the main cause of a worsening in the relationship between Afghanistan and the US.
US President Barack Obama has also apologised for the killings, which he described as a "tragic accident".
The boys, who had been gathering firewood, were mistaken for insurgents and killed by US helicopter gunships.
Hundreds of Afghans rallied on Sunday to denounce the killings. The protesters condemned both Nato and the Taliban for killing civilians.
Civilian casualties
Mr Gates told troops at Bagram airbase that both the US and Afghan governments agree that the US should remain involved even after 2014, when troops withdrawal is scheduled to be completed.
"Obviously it would be a small fraction of the presence that we have today, but I think we're willing to do that," he is quoted by the Associated Press news agency as saying.
While many Afghans accept that American troops are needed to defeat the Taliban, they resent their presence in the country, says the BBC's Quentin Sommerville in Kabul.
The war is in its 10th year, civilian casualties are at an all-time high, and the population has grown weary of the fighting.
Insurgents are to blame for most of the deaths, but killings by foreign troops generate widespread outrage, our correspondent adds.
As well as meeting President Karzai, Mr Gates will travel to the south and east of the country, on what is his 13th visit to Afghanistan.
"This is not a decision-making trip," Geoff Morrell, Mr Gates's press secretary, is quoted as saying by the AFP news agency.
"We are going to go south, we are going to go east, and he will come away from this visit hopefully with a better sense of how far we've come in the past three months," he said.
The Obama administration has said it will begin withdrawing US troops from Afghanistan in July and the exercise will be complete by 2014.
But Mr Gates is yet to indicate how many of the 97,000 US forces in the country will be withdrawn.
