Both companies have approached programmers about a possible agreement as they battle each other to deliver streaming video to their users, the report says.
According to people who have spoken to the Post and have seen the pitches, Facebook is already talking to the U.S. NFL for the Thursday Night Football package and is discussing offering a series of channels to its 1 billion-plus users.
Twitter, meanwhile, has been pushing its ability to help bring younger viewers to TV networks.
"All of a sudden, Facebook and Twitter are trying to get in the tent with us," one source told the Post. "They're each arguing their attributes and why aligning with them is to our advantage."
CNBC has contacted both Twitter but has yet to receive comment.
Facebook contacted CNBC after this story was published and said the company is "not focused on acquiring the rights to conventional TV programs."
UPDATE: This story has been updated with comments from Facebook.