"Mansour was seriously injured. He was rushed to hospital and we are not sure if he survived his wounds," Sultan Faizi, the spokesman for the Afghan first vice president, told AFP.
An Afghan intelligence source confirmed his account, and insurgent sources also told AFP that Mansour had been injured.
Taliban spokesman Zabihullah Mujahid denied that any such firefight took place.
Mansour was announced as the new Taliban chief on July 31, after the militant movement belatedly confirmed the death of Mullah Omar, who led the Islamist movement for some two decades.
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Four people were killed in the firefight today, a Taliban source close to Mansour's group told AFP, and a number of people injured.
"Akhtar Mansour is among the injured but the extent of his injuries is not clear," he said.
His account was confirmed by another Afghan Taliban source.
The sources differed on the location of the confrontation, with the insurgents saying it took place near Kandahar, the main city of southern Afghanistan and the birthplace of the Taliban, while other sources said it was in the Pakistani city of Quetta, a historic Taliban refuge.
The Taliban sources and a Western source said Sarhadi was a former Guantanamo Bay prisoner.
"There were differences on some points which later turned into harsh words, then Sarhadi opened fire and the others returned fire," the Taliban source said.
The news of Mansour's injury comes amid renewed official efforts to revive peace talks with Taliban insurgents.
The talks stalled after the Taliban belatedly confirmed the death of Mullah Omar.
The United States and China have been pushing for the process to restart, but frosty ties between Islamabad and Kabul have been hampering those efforts.
But despite the divisions, there has been no let up in insurgent attacks -- and the Taliban has seen a new resurgence under Mansour.
The Taliban briefly captured the strategic northern city of Kunduz in September in their most spectacular victory in 14 years of war.