Under the terms of the deal, the buyers’ consortium, which includes investment firm Silver Lake and Microsoft, will pay $13.65 a share in cash. That is roughly 25 percent above where Dell’s stock traded before word emerged of the negotiations of its sale.

Michael S. Dell will contribute his stake of roughly 14 percent toward the transaction and will contribute additional cash through his private investment firm, MSD Capital. Silver Lake is expected to contribute about $1 billion in cash, while Microsoft will loan an additional $2 billion.

Dell’s board is said to have met Monday night to vote on the deal. In its statement, the company said Michael Dell recused himself from any discussions about a transaction and did not vote.

As a newly private company — now more firmly under the control of Dell — the computer maker will seek to revive itself after years of decline. The takeover represents Michael Dell’s most drastic effort to turn around the company he founded in a college dorm room in 1984 and expanded into one of the world’s biggest sellers of personal computers.

But the advent of new competition, first from other PC manufacturers and then from smartphones and tablets, severely eroded Dell’s business. Such is the concern about the company’s future that Microsoft agreed to lend some of its considerable financial muscle to shore up one of its most important business partners.

“I believe this transaction will open an exciting new chapter for Dell, our customers and team members,” Michael Dell said in a statement. “Dell has made solid progress executing this strategy over the past four years, but we recognize that it will 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.”