Mullah Abdul Rauf said Mansour died in the strike late on Friday night “in the Afghanistan-Pakistan border area”.
The office of Afghan president Ashraf Ghani confirmed the strike, but could not confirm Mansour’s death.
However, chief executive Abdullah Abdullah said that Mansour is “more than likely” to be dead.
Mansour led the Taleban after the death of the movement’s founder, Mullah Mohammad Omar, was announced last summer.
Mansour, who was Mullah Omar’s deputy, concealed his superior’s death for more than two years, and ran the Taleban in his name until the death was disclosed by the Afghan government.
Speaking live on television as he chaired a cabinet meeting, Mr Abdullah said Mansour’s death would have a positive impact on attempts to bring peace to Afghanistan, where the Taleban have been waging an insurgency for 15 years.
Mansour was “the main figure preventing the Taleban joining the peace process”, Mr Abdullah said.
“From the day he took over the Taleban following the death of Mullah Omar, he intensified violence against ordinary citizens, especially in Afghanistan.”
Mullah Rauf was an early detractor of Mansour’s but decided this year to declare loyalty to him in the interest of unifying the movement.
Earlier, the US Department of Defence said a drone strike had targeted Mansour “in a remote area of the Afghanistan-Pakistan border region”.
Afghan officials said the drone strike took place in Pakistan’s Baluchistan province, in the Ahmad Wal area.
The Afghan government has long accused the Pakistani authorities of harbouring and supporting the Afghan Taleban.
The drone strike targeted Mansour’s vehicle which was carrying Mansour and one other person at the time, a US military source said.
Another Taleban source identified the driver as Muhammad Azam Hasanai, and said the vehicle the two men were travelling in was completely destroyed in the drone strike.
In Afghanistan, the National Directorate of Security, as the secret service is known, said in a statement that Mansour was killed in Baluchistan province, in south-western Pakistan.
“The attack happened on the main road while he was in his vehicle,” it said. It referred to “others” in the vehicle without giving any further details.
Since the 2001 ousting of Mullah Omar and the Taleban by a US-led military coalition, the Taleban founder was seen only by a handful of followers, including Mansour.
This recently caused some people within the movement, including Aga, to speculate that Mansour had intentionally concealed Mullah Omar’s death from the rest of the movement. Mansour was openly accused by several within the Taleban of manufacturing recordings by Mullah Omar and issuing orders using his name.
When the Taleban ruled Afghanistan between 1996 and 2001, Mansour held several positions in the government. In 1999 when the United Nations imposed wide-ranging sanctions on the movement, Mansour was both aviation minister, trying to keep the national airline operating without spare parts, and military commander in the Taleban stronghold of Kandahar.
Among the Taleban leaders at the time, Mansour was considered among those closest to Mullah Omar and, like the Taleban founder, Mansour rarely travelled to Kabul – instead running his aviation ministry from the southern city of Kandahar.