Starz's shares rose 8 percent to $41.89 in extended trading. The company had a market value of about $4 billion as of Tuesday close.
AMC's spokeswoman Georgia Juvelis said the company does not comment on rumors. A Starz spokesperson also said the company does not comment on rumors or speculation.
Starz ended talks for a buyout last year with potential buyers, including AMC, due to differences over valuation, Bloomberg reported in December. (http://bloom.bg/1LwDFi9)
Malone is the largest individual investor in Starz, which makes TV shows such as "Outlander" and "Da Vinci's Demons." The company was spun off from Malone's Liberty Media.
Shares of AMC, known for shows such as "The Walking Dead," were slightly up at $74.44 in extended trading after closing 2.4 percent higher in regular trade.