The Post cited two unnamed intelligence officials, who said forensic experts confirmed it while combing through the aftermath wreckage at the building of a Paris suburb after a 7-hour siege. Reuters, which cited the Post report, said it could not independently confirm it.
"I have information that led us to suspect that he has committed suicide. I do not have any particular documents saying that but that is the first indication we have received," Ambassador Francois Richier said. "We need to wait for confirmation of the outcome of the operation."
A DNA test is expected to provide the final evidence.
Abaaoud, the 28-year-old member of the Islamic State terror group, was believed to be among the suspects who were holed up in an apartment during the seven-hour raid by the French police and army.
Abaaoud was initially believed to be in Syria.
Two terrorists were killed, including a woman who detonated a suicide belt. The woman, say unconfirmed reports, was Abaaoud's relative.
Abaaoud was thought to have pulled the strings from Syria for the attacks that killed 129 people in Paris on Friday.
Some residents were evacuated, some still in their underwear, and authorities warned residents to keep away from windows.
The raid came as Europe was placed on high alert after footage from the scene of one of Friday's attacks in Paris, which killed 129 and injured 350, revealed a ninth suspect may have taken part.
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