Icahn met with Cook in late September to discuss the potential of a bigger buyback, which he called a "no brainer."
(Read more: A lesson in Icahn-Netflix trade of when to get out)
Icahn said he pressed Cook for a $150 billion share buyback, however, the two came to no agreement during their discussion. Since then, Icahn has made it very clear that he has no intention of giving up on a bigger Apple share buyback.
(Read more: Icahn: I pushed for a $150 billion buyback from Apple CEO Tim Cook )
"I feel very strongly about this," Icahn told CNBC after his meeting with Cook. "I can't promise you the stock will go up and I can't promise you they will do the buyback. But I can promise you that I'm not going away until they hear a lot more from me concerning this."
Former Apple CEO John Sculley weighed in on the situation Wednesday when he told CNBC that he doesn't think Cook should succumb to Icahn's push for increased dividends and a larger buyback.
"Well, Carl Icahn is one smart guy," he said. "But the reality is Apple is about building great products, great experiences, and if I were Tim Cook, I'd deal with him the way Steve Jobs would probably with him—not blink."
This story is developing. Please check back for further updates.
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--CNBC's Scott Wapner contributed to this report