(CNN) -- Afghan President Hamid Karzai and U.S. Defense Secretary Chuck Hagel met over dinner in the Afghan capital on Sunday in an attempt to smooth over the latest dispute in the already strained relationship between the two allies.
Hagel told reporters he tried to reassure Karzai that the United States had no unilateral back-channel talks with the Taliban and said Washington is still on track to wind up its 11-year combat mission in Afghanistan by the end of 2014.
"The fact is, any prospect for peace or political settlements -- that has to be led by the Afghans. That has to come from the Afghan side," Hagel said. "Obviously, the United States will support efforts if they are led by the Afghans to come to some possible resolution."
Hagel, a former senator who took the helm at the Pentagon last month, is making his first trip to Afghanistan as defense secretary. Karzai, meanwhile, has been increasingly critical of American forces in recent months.
Sunday, after a weekend bombing in Kabul that killed at least nine people, he said there are "ongoing daily talks between Taliban, American and foreigners in Europe and in the Gulf states" and that Saturday's attack shows "that Taliban want longer presence of foreigners -- not their departure from Afghanistan."
While there have been reconciliation talks in the past, the United States has not said any such talks are currently under way. In January, a U.S. official said reconciliation talks were showing "some signs of life" after being dormant for a year.
Hagel's dinner with the Afghan leader in Kabul came after a scheduled joint news conference between the two was canceled. Pentagon spokesman George Little said the schedule had changed "for a variety of reasons, to include decisions related to security in Kabul that were reached in consultation with our Afghan partners."
"I know these are difficult issues for President Karzai and the Afghan people," Hagel said after the meeting. "I was once a politician, so I can understand the kind of pressures that especially leaders of countries are always under. I would hope, again, that we can move forward, and I have confidence that we can and will deal with these issues."
Tensions as U.S. plans troop pullout
The Taliban claimed responsibility for Saturday's attack at the Afghan Ministry of Defense in Kabul, which killed at least nine people and wounded 14 others. A Taliban spokesman expressed pleasure with Hagel's proximity at the time, calling the attack "a message to him."
NATO's International Security Assistance Force rejected suggestions that the Taliban even knew of Hagel's trip when they planned the operation.
Afghanistan's National Security Council, chaired by Karzai, recently accused "armed individuals named as U.S. special force" of torturing and murdering innocent people in Wardak province. The government demanded members of the elite U.S. military units leave the province west of Kabul.
But the council also said the United States rejected such suggestions. U.S. military officials said all allegations of misconduct are taken seriously, and that the military was looking into the allegation.
On Sunday, senior ISAF officials were continuing talks with Karzai over Wardak, a province strategically located along a main transit route for insurgents attempting to reach Kabul.
Last year, Karzai called for U.S. troops to pull out of outposts in Afghan villages and return to their main bases.
In October, he complained the United States was failing to supply Afghan forces with weapons needed to fight insurgents.
Then-U.S. Defense Secretary Leon Panetta responded at the time that "it would be helpful if (Karzai), every once in a while, expressed his thanks for the sacrifices that have been made by those who have fought and died for Afghanistan, rather than criticizing them."
The Obama administration is making decisions about bringing troops home.
In January, Panetta reiterated his opposition to pulling out all U.S. troops by 2014, saying it would take away negotiating leverage with the Taliban.
"The stronger position we take about staying, the better chances we have to ultimately reach political reconciliation," Panetta told journalists.
Hagel met with U.S. and Afghan troops in Afghanistan on Sunday, thanking them for their service.
CNN's Josh Levs reported from Atlanta. CNN's Barbara Starr contributed to this report.