Algeria-born Abu Zeid was last seen in public in Timbuktu and Gao last year
A senior al-Qaeda militant has been killed in northern Mali, Chadian President Idriss Deby has said.
He said the country's forces killed Abdelhamid Abou Zeid during clashes in the remote region.
Algerian security services are said to have taken DNA samples from two of Abou Zeid's relatives to compare with the body which is reportedly his.
He is said to be second-in-command of al-Qaeda in the Islamic Maghreb, which is fighting foreign forces in Mali.
"Chadian forces killed two jihadi leaders, including Abou Zeid," President Deby said on Friday, without giving any further details.
'Most violent commander'
He was speaking after funerals of Chadian soldiers killed in the fighting in Mali.
Separately, reports in Algerian media said that the authorities were comparing DNA from Abou Zeid's relatives to confirm the militant's death.
A US official - speaking on condition of anonymity - said Washington found reports that Abou Zeid was killed "very credible", according to the AFP news agency.
However, France reacted with caution to the reports, with government spokeswoman Najat Vallaud-Belkacem stressing that his death was so far unconfirmed.
Earlier unverified reports in the French media said that the militant was killed during fighting against French army units.
Algeria-born Abou Zeid - believed to be in his 40s - was known as the most violent al-Qaeda commander in the region. He was last seen in public in Mali's cities of Timbuktu and Gao seized by Islamist groups last year.
Malian troops - together with French and African forces - have since recaptured most of the previously rebel-held territory.
Abou Zeid is also blamed for kidnapping dozens of Westerners in the vast region in recent years and killing at least two of them.
If confirmed, his death will immediately raise questions over the state of several French hostages who are widely believed to have remained in Abou Zeid's custody to date, the BBC's Thomas Fessy reports.