KABUL, Afghanistan - Negotiators from the United States and the Taliban are close to a deal that would open the way for peace in Afghanistan, a top US official said on Sunday, as the insurgents followed their weekend assault on the strategic center of Kunduz by attacking a second northern city.

Zalmay Khalilzad, the Afghanborn US diplomat overseeing negotiations for Washington, has met with the Afghan president in the capital, Kabul, to brief him on the latest round of talks with the Taliban as a deal nears on ending America's longest war, an official said on Monday.

Afghan President Ashraf Ghani's spokesman Sediq Sediqqi confirmed the meeting took place on Sunday night at the presidential palace, shortly after the US envoy arrived from Qatar.

In Qatar, Khalilzad had held the ninth round of talks with the Taliban, which ended without a final agreement. The palace soon will release details, Sediqqi said.

Khalilzad over the weekend said the US and the insurgent group are "at the threshold of an agreement" - even as the Taliban attacked the capitals of Kunduz and Baghlan provinces in the north.

"We are on the verge of ending the invasion and reaching a peaceful solution for Afghanistan," said the Taliban spokesman in Qatar, Suhail Shaheen.

The Taliban have stepped up attacks in recent months to strengthen their negotiating position while the United Nations and others say civilians have suffered, often caught in the cross-fire as government forces, backed by the US, have pursued the militants with airstrikes and raids. Afghanistan was the site of world's deadliest conflict in 2018.

A US-Taliban agreement on ending nearly 18 years of fighting is expected to be followed by intra-Afghan talks that include the Afghan government, which so far has been sidelined from the negotiations. The Taliban have refused to talk with the Afghan government, calling it a US puppet, but have expressed openness to speaking with Afghan officials in their personal capacity.

Some Taliban officials have said they would only agree to talk to Afghan officials in a private capacity, not as representatives of the state, and they remain opposed to presidential elections scheduled for Sept 28.

The insurgent group is at its strongest since the US-led invasion to topple its government after the Sept 11, 2001 attacks on the US. The Taliban now control or hold sway over roughly half of Afghanistan.

The Taliban want all of the estimated 20,000 US and NATO forces to leave the country and already portray their departure as the insurgents' victory. For its part, the US seeks Taliban assurances that Afghanistan will not be a safe haven for extremist groups to plan and launch global terror attacks. A cease-fire also has been discussed.

Few details have emerged from this latest round of US-Taliban peace talks, adding to the uncertainty as violence increases.

The agreement with the Taliban "will reduce violence and open the door for Afghans to sit together to negotiate an honorable& sustainable peace and a unified, sovereign Afghanistan that does not threaten the United States, its allies, or any other country", Khalilzad said on Twitter before his Kabul arrival.