Icahn met with Cook in late September to discuss such a buyback. However, at the time Cook did not seem interested in a buyback of that size, Icahn told CNBC.
In late October, Icahn put more pressure on Cook by publishing an open letter on his new website called "Shareholder's Square Table."
(Read more: Carl Icahn's letter to Tim Cook )
Shortly after publishing the letter, Icahn told CNBC that he had no intention of giving up on a large buyback and said he may consider a proxy battle to return more value to shareholders.
(Read more: Carl Icahn: Could consider an Apple proxy fight )
"If they don't do it, we'll test the waters and see if the shareholders want us to do it and if we could win," Icahn told CNBC. "We've said we're not going away. I've said that to (CEO) Tim (Cook) and I'll say it again."
Icahn said in his letter that he owned about $2.5 billion worth of Apple's stock and said that he intends to continue to grow that position.
(Read more: Icahn, Transocean reach deal to resolve proxy battle)
Click here for the latest on the markets.