"At this time, it looks like we will not reach an agreement and our channels may very well go off the air," Carey wrote in the memo.
Carey added that any extension to the negotiations simply would give Time Warner more cheap access to Fox shows.
"The fact is we've been trying since the summer to negotiate a fair deal and that further extensions simply extend the period of time that Time Warner profits from our marquee programming," Carey wrote.
The public spat began in November, when Time Warner Cable launched an ad campaign airing its grievances against programmers and asked viewers to vote on whether Time Warner should "roll over" and cough up the higher fees, or whether it should "get tough."
This week, Time Warner Cable started running dramatic ads in ransom-style lettering in newspapers around the country.
"Pay our price or you'll never see Fox again," reads a Time Warner Cable ad.
Fox responded with ads of its own: "No Fox? No Way," read News Corp.'s ads over a collage of pictures featuring its hit shows, including "NFL on Fox," "American Idol," "24," "House" and "The Simpsons."
If Fox pulls its content in January, it wouldn't be the first time a programmer used viewers -- who usually direct their anger at their cable company, not the content provider -- as negotiating pawns.
"It's an ongoing war," said Carl Howe, director of media consumer research at the Yankee Group. "Distributors want as much of the money as possible, and the people who create the content want their share."
The dispute came at a difficult time for both companies. Cable operators such as Time Warner Cable are beginning to lose leverage with their customers because more Americans are getting their content for free on Internet sites such as Hulu.
Programmers like Fox, meanwhile, have seen advertising revenues drop because of changes in the advertising industry and the recession.
To replace lost income, programmers have begun asking cable distributors for a higher cut of subscription revenues.
Time Warner Cable, which had several contracts in addition to the Fox deal expiring at the end of the year, said some programmers have asked for fee increases of up to 300 percent.
"When a programmer comes to us and asks us for a 300 percent price increase for their content, that's what causes your cable bill to go up," said Time Warner Cable's Dudley. Programmers, he argued, already get their fair cut.
Time Warner Cable recently split off from its former parent, Time Warner Inc., which owns Time magazine, CNN and Warner Brothers.
Last year, Time Warner received $16.3 billion from subscribers, and paid $3.7 billion -- almost a quarter of that -- to content providers such as Fox.
While that seems like a small percentage, cable operators point out that maintaining a national network to deliver content is costly.
ABC News' Michael S. James contributed to this report.