If it's not settled, six CBS TV stations could go dark for around 3 million Time Warner Cable customers in New York, Los Angeles and Dallas.
CBS Inc. and Time Warner Cable Inc. are in the public spat over fees in the markets.
Moonves told the Television Critics Association the deadline to resolve the dispute was 5 p.m. EDT Monday. Later, both companies said they extended the deadline to 8 p.m. EDT.
The disagreement centres mainly on how much Time Warner Cable pays for the right to retransmit signals from the CBS-owned stations.
Also involved is a possible blackout of CBS-owned Showtime for Time Warner Cable customers nationwide who pay extra for it.
Discussions were continuing and Moonves said he hopes the stations don't go dark.
"As we've said, we feel like we should be paid for our programming," he said. He declined to offer further details, saying he didn't want to negotiate in public.
Dozens of blackouts have occurred nationwide in fee fights over the years, but many get resolved at the last minute.
Selling retransmission rights has become a big business for broadcasters such as CBS. Research firm SNL Kagan estimates retransmission fees will reach $3 billion industrywide this year and double to $6 billion by 2018.
Time Warner Cable has said it's resisting a fee hike demand by CBS so prices don't go up for customers. CBS said Time Warner Cable isn't agreeing to terms that its competitors have accepted.
If the fight continues, some Time Warner Cable customers could lose access to new episodes of "Under the Dome" and the 15th season of "Big Brother." Both are available online after a slight delay.