Robert Gates, the US defence secretary, has arrived in Afghanistan at a time of increased strain between Kabul and its Western supporters with important security transition milestones looming.
His arrival on Monday marked the start of a two-day visit which had not been announced in advance.
US military Gates said on Monday that both the US and Afghan governments agree on the US military involvement in Afghanistan after the planned 2014 end of combat operations to help train and advise Afghan forces.
"Obviously it would be a small fraction of the presence that we have today, but I think we're willing to do that," Gates told a group of US troops at Bagram air field, US and NATO forces headquarter in eastern Afghanistan.
"My sense is, they (Afghan officials) are interested in having us do that."
Responding to a question about the possibility of a long-term military presence, Gates said that Washington and Kabul have recently begun negotiating a security partnership, however he did not give any details.
Earlier, Geoff Morrell, US defence department spokesman, told reporters that Gates wants to get a first-hand feel for changes on the ground since he was last in Afghanistan in December.
Gates is due to meet Hamid Karzai one day after the Afghan president said an apology from General David Petraeus, the commander of US and NATO forces in the country, over a string of recent accidental civilian killings was "not enough".
The issue of civilian casualties has clouded the relationship between Kabul and its allies, taking attention away from transition plans.
Karzai will soon unveil a timetable for the beginning of the gradual handover of security responsibility from foreign forces to Afghans. The process is to begin in July to be completed by 2014.
Karzai complained angrily last week after nine Afghan children were mistakenly killed by helicopters from the NATO-led force. The boys were gunned down while collecting firewood in a volatile eastern province last Tuesday.
US President Barack Obama has expressed his "deep regret" over the incident.
At a meeting with security advisers on the eve of Gates' trip, at which Petraeus was present, Karzai said civilian casualties caused by foreign troops were "no longer acceptable".
During the meeting, Petraeus again apologised for the killings, saying they were a "great mistake", according to a statement released by the presidential palace.
Earlier on Sunday, hundreds of Afghans chanting "Death to America" gathered in the capital to protest against such casualties. There have been at least four incidents involving civilian casualties, mainly in the east, in the past three weeks.
International concern over civilian casualties has grown, and the fallout from the recent incidents is threatening to hamper peace and reconciliation efforts ahead of the gradual drawdown of some 150,000 foreign troops still deployed in the country.
This week's visit is Gates' 13th trip to Afghanistan, and probably one of his last as defence secretary. He has said he will retire this year but has not given a date yet.
US envoy in Pakistan
Meanwhile, Marc Grossman, the new US special envoy to Afghanistan and Pakistan, met with top officials in Islamabad, on Monday, during his first trip to the region since taking the position previously held by the late Richard Holbrooke.
Grossman's visit comes with US-Pakistani relations at a low point due to the dispute over the detention of Raymond Davis, an American CIA contractor who shot dead two Pakistanis.
The Davis case has been a major setback to cooperation between Pakistan and the US, and it is likely to figure in Grossman's talks with Pakistani leaders.
Washington says that Davis has diplomatic immunity and acted in self-defence against robbers. However, a Pakistani court has ruled that he should stand trial as h