Specifically the Afghan Taliban praised India for sending U.S. Defense Secretary Leon Panetta "empty-handed toward Kabul," a report by the Press Trust of India in the Times of India newspaper said.
The Times quoted from a Taliban statement issued in Kabul which said Panetta "spent three days in India to transfer the heavy burden to (India's) shoulders, to find an exit and to flee from Afghanistan."
The statement reportedly said India is "aware of the Afghans' aspirations, creeds and love for freedom. It is totally illogical they should plunge their nation into a calamity just for American pleasure."
The Taliban said "Indian authorities didn't pay heed to (U.S.) demands and showed their reservations" because New Delhi understands the United States is digging its own grave.
"Indian people and their authorities are observing this illicit war for the last 12 years and they are aware of the Afghan nation and their demands," the statement said.
India traditionally has had good relations with Afghanistan, including during the days of Soviet-backed government in Kabul.
Afghanistan governments have seen relations with India as a counterweight to its often tumultuous relationship with Pakistan, especially over border security issues.
Afghan President Hamid Karzai has made several trips to India since taking up the reins of power in 2001.
During Karzai's visit to New Delhi in 2006, Indian Prime Minister Manmohan Singh pledged $50 million more in assistance to Afghanistan. The amount took India's pledge of aid to Kabul to around $650 million.
India was also reconstructing a road in the remote southwestern Afghan province of Nimroz, Radio Free Europe said.
India and Afghanistan also have signed preferential trade agreements for specific items. Afghanistan has cut duty on Indian black tea, cement products and medicines for retail sale. India has lowered tariffs on many of Afghanistan's dried fruit goods as well as some gem stones, including rubies and emeralds.
But trade between the countries has been hampered by the logistics of trucking Indian goods to Afghanistan through the territory of Pakistan.
India and Pakistan remain deadlocked over border and territorial issues, including Islamabad's dispute over India's Kashmir area.
But during a 2006 trip to Pakistan, Karzai said Afghanistan's "relations with India in no way will impact" ties between Kabul and Islamabad.
Relations between New Delhi and Kabul were shaken in July 2008 in the aftermath of a suicide bomber attack on India's embassy that killed 58 people and wounded 141 others. The suicide car bombing happened close to the embassy gates during the morning rush hour.
Suspicion for the blast fell on Pakistan, a report by the New York Times said less than a month after the attack.
U.S. intelligence agencies concluded that members of Pakistan's spy service were involved in the attack, The New York Times report said, citing U.S. government officials as its source.