Both companies have approached programmers about a deal, the Post said, citing several sources familiar with the situation.
Facebook, which is already in talks with the National Football League for digital rights to Thursday Night Football, has met with a wide range of TV executives over the past few weeks, sources told the Post.
However, it was not known how Facebook would deliver the shows, sources who heard the pitch told the newspaper.
“Our goal with live video is to work with our partners to move to a sustainable monetization model quickly,” Facebook said in a statement.
“We are not focused on acquiring the rights to conventional TV programs.”
Twitter declined to comment.