A report in the American newspaper quoted unnamed US officials as confirming the meeting.
If confirmed, this would be the first highest-level interaction between Taliban and the US government since the insurgent group took over Kabul. The development comes at a time when thousands of people from Taliban-held Afghanistan are seeking evacuation.
William Burns is a senior US diplomat and Mullah Baradar, who was the head of the Taliban's Qatar office, is among the top Taliban leaders who are now handling the affairs in Kabul as the country's de facto rulers.
AFP reported that a CIA spokesman refused to confirm the meeting, saying that the agency does not talk about the director's travels.
While the WashPost report did cite US officials, it did not mention as to what kind of discussions took place between the Taliban leader and the CIA chief. The report however says that the possible delay in the deadline for the US to complete evacuations at the Kabul airport was likely addressed during the meeting.
Thousands of Afghan citizens, who do not wish to live under the Taliban rule, are gathered at the Kabul Airport in hopes of fleeing the country.