They say he was killed in a drone strike carried out by the CIA in the Afghan capital, Kabul, on Sunday.
There is no independent confirmation of this at present.
However, the White House says that President Joe Biden will give details of the operation in Afghanistan in the next few hours.
CBS News said three sources had confirmed the killing of the al-Qaeda leader. The New York Times, Washington Post and CNN also quoted unidentified sources identifying the victim.
A senior US official earlier confirmed there had been a “successful” operation against a “significant” al-Qaeda target in Afghanistan.
The operation took place over the weekend and there were no civilian casualties, the official added.
At the same time, a Taliban spokesman reported that a US drone strike had taken place on Sunday in a residential area of Kabul.
He described it as a clear violation of international principles.
“Such actions are a repetition of the failed experiences of the past 20 years and are against the interests of the United States of America, Afghanistan and the region,” the spokesman added.
Zawahiri, an eye surgeon who helped found the Egyptian Islamic Jihad militant group, took over the leadership of al-Qaeda following the killing by US forces of Osama Bin Laden in May 2011.
Before that, Zawahiri was often referred to as Bin Laden’s right-hand man and the chief ideologue of al-Qaeda.
He is believed by some experts to have been the “operational brains” behind the 11 September 2001 attacks in the United States.
Ayman al-Zawahiri was the ideological brains behind al-Qaeda.
An Egyptian doctor who was imprisoned in the 1980s for involvement in militant Islam, he left the country after his release and became involved in violent international jihadist movements.
Eventually he settled in Afghanistan and joined forces with a rich Saudi, Osama Bin Laden. Together they declared war on the US and organised the 11 September 2001 attacks.
It took a decade for Bin Laden to be tracked down and killed by the US. After that, Zawahiri assumed leadership of al-Qaeda, but he became a remote and marginal figure, only occasionally issuing messages.
If his death is confirmed, the US will herald it as a victory, particularly after the chaotic withdrawal from Afghanistan last year, but Zawahiri held relatively little sway as new groups and movements such as Islamic State have become increasingly influential. A new al-Qaeda leader will no doubt emerge, but he will likely have even less influence than his predecessor.
