Afghan-born Khalilzad, a former USA ambassador to Kabul, Baghdad and the United Nations, was appointed to steer peace efforts with the Taliban last month.
The Taliban had reportedly met with USA officials in July after an unprecedented ceasefire in June, which fuelled hopes that talks could bring an end to fighting after 17 years.
But under pressure to break the impasse with the group, Washington appeared to change tack in June when US Secretary of State Mike Pompeo said his country was prepared to "participate" in talks.
It went on to say that the presence of "foreign forces" in Afghanistan was the biggest concern of the Taliban.More news: President Trump calls for 'severe punishment' if Saudi Arabia killed US-journalist
A statement from the Afghan president's office on October 13 said Khalilzad told Ghani and Abdullah that the United States was "ready to do anything to help with the peace process", but insisted the process should be led by the Afghan government.
The Governor of Pusht Rod district confirmed the attacks by the Taliban on two military check posts and told Anadolu Agency that the confrontation started on Saturday night and carried on till the wee hours of Sunday morning.
In the western province of Farah, Taliban fighters killed 21 troops at two checkpoints in the district of Posht-e Rud.
Talks with the Taliban have since been taking place elsewhere in Doha.
And in the western province of Herat, gunmen attacked the campaign office of a candidate, killing a security guard and a child. He was holding a meeting at the time of the attack in Lashkar Gah, the provincial capital.More news: No deal yet after flurry of Brexit diplomacy
At least 68 people were killed when a suicide bomber detonated explosives at an election rally. There are now about 14,000 United States soldiers in the country.
There are 2,565 candidates vying for seats in the 249-member chamber, including 417 women candidates. Afghan security forces accidentally killed three bodyguards of an independent candidate during a raid on a house near his residence in the eastern Kunar province.
The two officials discussed ways of achieving peace and stability in Afghanistan and Qatar's role in the peace process in the country, The Peninsula Qatar reported.
At least eight Afghan police personnel, including a district police chief, were killed in clashes in southern province of Zabul overnight, a provincial source said Sunday.
The hardline Islamist terrorist group claimed the attacks.More news: Call of Duty: Black Ops 4 breaks day one digital sales records