Icahn revealed in August that he had purchased about $1 billion worth of Apple's shares.
Earlier this year, Apple announced that it had increased its dividend and stock buyback program to more than $100 billion, including $60 billion to buy back shares of the company's stock.
Icahn said in his tweet Tuesday that he wants to see the buyback portion increased even further:
Had a cordial dinner with Tim last night. We pushed hard for a 150 billion buyback. We decided to continue dialogue in about three weeks.— Carl Icahn (@Carl_C_Icahn) October 1, 2013
After the dinner, Icahn said the bigger buyback makes perfect sense.
"We argued very hard for it, we think it’s a no-brainer," Icahn told Bloomberg.
Bloomberg also reported that Cook dined at Icahn's New York apartment, the first time the pair had met in person.
Thus far, Cook has not reciprocated on his Twitter account.
Later, in an interview on CNBC, Icahn expanded on his rationale for the increased buyback.
He said Apple's stock is relatively cheap at the moment, and the cost of borrowing money is low. Taking more shares off the market could increase the share price significantly, he said.
"I feel very strongly about this," Icahn said on CNBC. "I can't promise you the stock will go up and I can't promise you they will do the buyback."
Icahn added: "I'm certainly not threatening anything. But on this issue, I feel very strongly and I'm not going to go away on it."
"It makes eminent sense," Icahn said. "You can borrow money so cheaply today. They can buy a stock that they also consider to be cheap. So throughout the evening I'm saying, 'Why wouldn't you do it? It makes no sense not to. I want a reason.' "
Finally: "The reason can't be the board decided," Icahn said. "That's complete bull as far as I'm concerned."
Following Icahn's remarks, Apple's stock was up $10.56 or 2.21% to $487.31.