"We see no reason that the future value of Dell should not accrue to ALL the existing Dell shareholders — not just Michael Dell," Icahn wrote in the letter, which Dell disclosed in a filing.
Icahn's proposed special dividend of $9 per share would include $4.26 a share derived from $5.25 billion in new debt.
He argued that, combined with a "stub" value of $13.81 per share that his firm calculated based on discounted cash flow, such a move would deliver shareholders a total value of $22.81 per share.
(Read More: Dell 'Welcomes' Carl Icahn to Go-Shop Process)
That would be a 67 percent premium to the $13.65 buyout price put forward by Michael Dell in partnership with private equity house Silver Lake and Microsoft.
In his letter, Icahn asked Dell's board to commit to his proposal to "avoid a proxy fight." He also wants Dell to combine an upcoming special shareholder vote on the buyout deal with its regular annual general meeting, when Icahn said he intended to put forward a slate of his own directors for election to the board.
"If you fail to agree promptly to combine the vote ... we anticipate years of litigation will follow challenging the transaction and the actions of the directors that participated in it," Icahn said in the letter.
To speed up the payment of a special dividend, Icahn himself pledged to provide a $3.25 billion bridge loan to Dell, if necessary.
Icahn joins other major investors opposed to the deal. Excluding the activist investor, who has not disclosed his stake, shareholders representing almost 18 percent of Dell shares, led by Southeastern Asset Management with a stake of more than 8 percent including options, say they will vote against the buyout.
The likelihood of alternative proposals prompting Dell to sweeten his offer has increased since a handful of major shareholders spoke out against the deal, analysts said.
(Read More: Dell Buyout Has '50-50' Chance: Wilbur Ross)
"Recent events have raised the level of scrutiny on the situation, which could end up increasing the take-private price," Wells Fargo analyst Maynard Um wrote in a note.
Southeastern, run by activist investor Mason Hawkins, has said Dell could borrow money to make a major share repurchase or break up the company and sell units separately.
T. Rowe Price has also spoken out against the deal.
Icahn said only that his interests held a substantial stake in Dell. He did not disclose his exact shareholding.
CNBC, citing unnamed sources, reported on Wednesday that Icahn had accumulated about 6 percent of Dell's shares.
That would make him the third-largest shareholder, after Southeastern and Michael Dell himself.
Dell shares finshed 0.7 percent lower at $14.22 on the Nasdaq on Thursday. They had risen about 8 percent since trading closed on Feb. 4, the day before Dell announced its plan to go private.
'Value for All'
Jefferies & Co. analyst Peter Misek said most investors were likely to favor an increased bid, but a rival offer was unlikely.
"Icahn would likely be satisfied with a raised bid to $15, which we think would be higher than the potential stock price realized from a leveraged recap," Misek wrote in a note.
"While a break-up of Dell would increase the probability of a strategic investor buying a portion of the business, we still see this as unlikely due to the size of the PC business and Michael Dell's desire to lead a shift toward enterprise hardware/software."
Dell said it would welcome Icahn's participation in a "go-shop" process to find alternative proposals, which the company announced on Wednesday.
In addition to Icahn's proffered personal bridge loan, his Icahn Enterprises LP would also provide $2 billion of bridge financing for the "prompt payment" of the dividend, if his slate of directors was elected, Icahn said in the letter.
"This is far from over," said Anthony Michael Sabino, a professor in the law department at St. John's University's Peter J. Tobin College of Business. "Expect a lot of pulling and tugging between multiple rivals here, with Dell as the prize."