The newspaper says Mr Cosby made the deposition 10 years ago. A lawsuit was lodged by a former university worker in Philadelphia who claimed he drugged and molested her.
Mr Cosby, now 78, is facing a series of sexual assault accusations.
He has denied the claims.
Court papers unsealed earlier this month showed that he admitted giving drugs to women with the aim of having sex with them.
The New York Times says it has obtained the deposition on which those court papers were based, in which more details are given.
Read More: bbc.co.uk
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