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.
Mr Cosby was questioned for four days in 2005 and 2006 in a Philadelphia hotel, and the deposition of close to 1,000 pages was formed.
The case was brought by Andrea Constand, a former employee of Temple University – the Philadelphia college with which Mr Cosby was once closely associated.
After Mr Cosby intiated what he called a mentoring relationship, Ms Constand says he then gave her a powerful drug and molested her.
Mr Cosby says he gave her only the over-the-counter antihistamine drug Benadryl.
‘These romantic sexual things’
After Ms Constand became upset and returned to her native Canada, Mr Cosby offered to pay for her further education.
“My wife would not know it was because Andrea and I had had sex and that Andrea was now very, very upset and that she decided that she would like to go to school,” the New York Times quotes Mr Cosby as saying in the deposition.
He is also reported as saying he was skilled in reading women’s reactions: “I think I’m a pretty decent reader of people and their emotions in these romantic sexual things, whatever you want to call them.”
Mr Cosby also admits paying another woman he met in 1976 and to obtaining sedatives with the aim of giving them to women to have sex with him.
And, still in the deposition, he says he asked a model about the cancer her late father suffered with the purpose of having sex with her.
Best known as Dr Cliff Huxtable on The Cosby Show in 1984-92, the comedian is facing a number of allegations dating back to the 1960s.
Most of the claims of sexual assault are barred by statutes of limitations. They restrict the length of time in which legal actions can be taken after an alleged crime has been committed.
The accusations, which Mr Cosby has described as “fantastical” and “uncorroborated”, have led to some of his stand-up shows being called off and the cancellation of some TV projects.