In a letter to Cook posted online on Thursday, Icahn said he has increased his stake in Apple by 22 percent from 3.9 million at the end of September. At Apple's current stock price, that's worth about $2.5 billion and amounts to less than a 1 percent stake in the company. He plans to increase his stake.
Icahn wants Apple to launch the buyback at its current stock price, which closed at close to $525 on Wednesday when the letter was sent. He stressed that he does not plan to tender any of his shares in the buyback he is proposing.
"There is nothing short term about my intentions here," he wrote.
A representative for Apple did not immediately respond to a message for comment.
Icahn said he wants to make it "very clear" that he supports Cook and Apple's current management team, as well as Apple's culture and "innovative spirit it engenders." It's just that he believes that Apple's current buyback plan - $60 billion over three years - is too small.
Speaking of his proposal, Icahn wrote, "While this would certainly be unprecedented because of its size, it is actually appropriate and manageable relative to the size and financial strength of your company. Apple generates more than enough cash flow to service this amount of debt and has $147 billion of cash in the bank."
The billionaire investor posted the letter on a new website he launched Thursday, called "Shareholders Square Table."
Shares of Cupertino, Calif.-based Apple rose $1.87 to $526.83 in morning trading.