BAGRAM AIR FIELD, Afghanistan — The United States accepts that a diminished but resilient Taliban is likely to remain a military threat in some parts of Afghanistan long after U.S. troops complete their combat mission next year, the top U.S. military officer says.

In an Associated Press interview at this air field north of Kabul, Army Gen. Martin Dempsey, chairman of the Joint Chiefs of Staff, said Sunday he is cautiously optimistic that the Afghan army will hold its own against the insurgency as Western troops pull back and Afghans assume the lead combat role. He said that by May or June, the Afghans will be in the lead throughout the country.

Asked whether some parts of the country will remain contested by the Taliban, he replied, “Yes, of course there will be.”

“And if we were having this conversation 10 years from now, I suspect there would (still) be contested areas because the history of Afghanistan suggests that there will always be contested areas,” he said.

He and other U.S. commanders have said that ultimately the Afghans must reach some sort of political accommodation with the insurgents, and that a reconciliation process needs to be led by Afghans, not Americans. Thus the No. 1 priority for the U.S. military in its final months of combat in Afghanistan is to do all that is possible to boost the strength and confidence of Afghan forces.

Shortly after Dempsey arrived in Afghanistan on Saturday, the Taliban demonstrated its ability to strike.

It claimed responsibility for a suicide car bombing that killed five Americans — three soldiers and two civilians, including Anne Smedinghoff, a foreign service officer and the first American diplomat killed overseas since the terrorist attack Sept. 11 in Benghazi, Libya.

A fierce battle between U.S.-backed Afghan forces and Taliban militants in a remote corner of eastern Afghanistan left nearly 20 people dead, including 11 Afghan children killed in an airstrike, Afghan officials said Sunday.

There are now about 66,000 U.S. troops in Afghanistan. That number is to drop to about 32,000 by February 2014, and the combat mission is to end in December 2014. Whether some number — perhaps 9,000 or 10,000 — remain into 2015 as military trainers and counterinsurgents is yet to be decided.

Dempsey spent two days talking to senior Afghan officials, including his counterpart, Gen. Sher Mohammad Karimi, as well as top U.S. and allied commanders.

He also visited a U.S. base in the volatile eastern province of Paktika for an update on how U.S. troops are balancing the twin missions of advising Afghan forces and withdrawing tons of U.S. equipment as the war effort winds down.

Paktika is an example of a sector of Afghanistan that is likely to face Taliban resistance for years to come.

Bordering areas of Pakistan that provide haven for the Taliban and its affiliated Haqqani network, Paktika has been among the more important insurgent avenues into the Afghan interior.

While the province has a functioning government, Taliban influence remains significant in less populated areas, as it has since U.S. forces first invaded the country more than 11 years ago.

“There will be contested areas, and it will be the Afghans’ choice whether to allow those contested areas to persist, or, when necessary, take action to exert themselves into those contested area,” he said.

Dempsey said he is encouraged by the recent development of coordination centers, including one in Paktika, where a wide range of Afghan government agencies work together on security issues. He called it a “quilt” of government structures that links Kabul, the capital, to ordinary Afghans in distant villages.

In some parts of the country, Afghan villagers have shown their dissatisfaction with Taliban influence by taking up arms against the insurgents, even without being pushed by the U.S. or by Kabul. This has happened in recent weeks in the Panjwai district of Kandahar province, a traditional stronghold of the Taliban. The Andar district of Ghazni province has seen a similar uprising.

“We should encourage it, but we shouldn’t be seen as hijacking” these local movements, he said.

Dempsey said he discussed the uprisings with Karimi, the army chief, and the Afghan defense minister, Bismullah Khan Mohammadi. They told him they “appreciated that they should allow this to occur (and) they should probably nurture it. They don’t necessarily feel at this point as if they should tangibly support it.”

The Afghan government’s concern, Dempsey said, is that influential warlords could embrace these local movements and eventually leverage them to threaten the armed forces of the central government.

In a separate interview Sunday with al-Hurra, the Arabic-language satellite TV channel funded by the U.S. government, Dempsey was asked whether he worries that Syria, in the midst of a civil war, could become another Afghanistan.

“I do. I have grave concerns that Syria could become an extended conflict” that drags on for many years, he said.