The proposed sale was expected to be announced Thursday.
If successful, the deal would create a single, huge video and Internet presence with 30 million subscribers and operations coast to coast.
Under terms of the agreement, Comcast, based in Philadelphia, would pay $158.82 a share for Time Warner Cable, offering 2.875 shares of its stock for each share of Time Warner Cable in the all-stock deal, the Times said.
Besides its cable and Internet operations, Comcast is the parent of NBCUniversal, which owns the NBC broadcast network, Universal Studios and cable channels MSNBC, USA Network and CNBC, which first broke the story Wednesday.
Time Warner Cable, which is headquartered in New York and also has been expanding into programming, owns two sports channels in Los Angeles.
The transaction must be approved by both Comcast's and Time Warner Cable's boards of directors. Time Warner Chief Executive Officer Rob Marcus, who took over the company's helm less than two months ago, will resign once the sale closes, the Times said.
In an early morning conference call, Marcus said it would be difficult to give up control of the company. However, the deal "just makes too much sense," he said.
"Much as I would have relished the opportunity to let my team seize the opportunity ahead, I believe that my customers and yours stand to benefit tremendously," he added.
Comcast CEO Brian Roberts said he has been considering the deal "for some time now," the Wall Street Journal reported.
The deal is "pro-consumer, pro-competitive and strongly in the public interest," he said, calling the deal "approvable."
The deal will not reduce competitive market advantages because they currently do not operate in the same ZIP codes, Robers said.
Charter Communications, another cable operator, last month offered $132.50 a share for Time Warner Cable, which was rejected.
The combination of Comcast and Time Warner Cable does not violate current Federal Communications Commission rules, the Times said, but likely will face scrutiny from regulators and lawmakers, many of whom have voiced concern about media consolidation and its effect on consumers.
Comcast officials said they would sell cable systems serving about 3 million subscribers so its national presence does not exceed 30 percent, sources told the Times. The 30 percent benchmark was the previous FCC cap on cable ownership until a federal court voided that rule in 2009.