Along with technology investor Silver Lake, and with financial backing from Microsoft, he will offer to buy the firm for $24.4bn (£15.5bn).
The move will take Dell off the Nasdaq stock exchange after 25 years.
Mr Dell hopes to turn the tide for a firm that has struggled to compete with cheap Asian rivals and the boom in smartphones and tablet computers.
Dell's success over the last 29 years has made its founder one of the richest men in America.
The chief executive and chairman already owns about 14% of the firm. He and fellow senior executives will retain their existing stakes.
The buy-out of the remaining shares will be carried out by a consortium made up of Mr Dell himself, his own investment fund, and Silver Lake. It will be financed by loans from four banks, and by a $2bn loan from Microsoft.
Analysts said the move would give Mr Dell greater flexibility in turning the company around, by dispensing with the need to deliver strong results every quarter to shareholders on the stock market.
The entrepreneur said that the firm's long-term strategy would "still take more time, investment and patience, and I believe our efforts will be better supported by partnering with Silver Lake in our shared vision".
By Rory Cellan-JonesTechnology correspondent
For consumers looking for a decent PC at a keen price, Dell was for many years the obvious choice.
Design was never the company's strong suit - even when the firm moved on from beige boxes - but for many years it vied with HP for the number one spot in the PC market.
Now though, according to one view, we are moving into the post-PC era, and that has left Dell struggling to look relevant as smartphones and tablets grab all the consumer attention.
The most recent figures from Gartner showed a 5% year-on-year  fall in global PC shipments in the last quarter of 2012 - but Dell's shipments fell by 21%.
So far, Dell's attempts to move into the tablet world have flopped, and the stock market has been impatient for results.
Michael Dell will hope that, away from the relentless spotlight of quarterly reporting, he can get on with the long-term task of transforming his business.
But, given the seismic changes taking place in the way we use computers, he may still have to move fast to make Dell matter to tomorrow's consumers.
"One of the key questions is going to be how much influence Microsoft is going to have over Dell's strategy," said Cindy Shaw, analyst at Discern Group.
Microsoft provides the Windows operating systems for Dell's computers, as well as for rival PC makers such as Hewlett Packard.
Michael Dell founded the company - originally named "PCs Limited" - in 1984 at the age of 19, operating out of his dorm room at the University of Texas.
He later dropped out of university in order to develop his PCs, including innovative laptops, and to expand his business overseas.
But over the last decade, Dell has faced increasingly steep competition, from cheaper Asian PC manufacturers such as China's Lenovo, and from innovative rivals such as Apple and Samsung whose tablets and smartphones have superseded the desktop and laptop.
The company appeared to lose its direction between 2004-07, when Mr Dell stepped down from day-to-day management of the company, and has suffered a number of false starts since his return, including the unsuccessful launch of its "Streak" tablet computer in 2010.
Mr Dell first approached the board with a buy-out proposal in August, and news of the talks emerged in mid-January, propelling the company's share price higher.
Together with Silver Lake, he is offering shareholders $13.65 in cash per share.
Dell's share price jumped a relatively modest 14 cents to $13.41 on Tuesday on Wall Street - reflecting the fact that the news was already well anticipated by the market.
The company's board pointed out the offer represented a 25% premium over Dell's share price in mid-January when rumours of the deal first broke.
However, the share price still remains some way below the $17-$18 level it was trading at a year ago, and well below its all-time high of just under $60 in 2000.
Completion of the buy-out, which will require shareholder approval, is far from agreed.
Before completing the deal, Dell's board said it will conduct what it called a "go shop" period of 45 days during which it would actively seek competing offers to buy the firm at a better price.
Michael Dell has said he will buy back the world's number three PC-maker that he founded and that carries his name.