The firms are now holding discussions on a deal with billionaire Chief Executive and founder Michael Dell, who owns about 14 percent of the company, according to one source with knowledge of the matter.
The Wall Street Journal cited unidentified sources as saying TPG and Silver Lake could team up on an offer, possibly in conjunction with other investors such as pension funds. JPMorgan Chase & Co was also involved in the negotiations, it added.
The first source told Reuters any potential deal could be structured as a management-led buyout with Michael Dell at the helm.
Talks had progressed for two to three months, heating up in late 2012, and a deal could be reached in six weeks, the Journal cited sources as saying.
Dell, which has steadily ceded market share to Hewlett Packard and China's Lenovo, declined to comment on what it called rumors and speculation.
The company has lost 40 percent of its value since last year's peak, and is trying to reinvent itself as a seller of higher-margin services to corporations - an internal overhaul that might be conducted away from public scrutiny.
Some analysts say taking the company private, an idea that has surfaced sporadically in past years, makes sense.
But others pointed to the sheer expense of such a deal, an outsized debt burden of some $4.5 billion and murky prospects as a major player in a PC market that's dwindling with the advent of tablets such as Apple Inc's iPad.
"The market value of Dell has come down so much that a buyout has become something that is plausible. They have about $5 billion in net cash and also free cash flow generation that could sustain payments on debt from a leveraged buyout," said S&P Capital IQ analyst Angelo Zino.
"However, we think it's unlikely, given the sheer size of Dell and where the stock is currently trading at."
A buyout of the $19 billion company would be one of the largest deals since the global recession.
Bloomberg first reported that Dell and private equity firms were discussing a deal.
Before news of the deal emerged, Sanford Bernstein analyst Toni Sacconaghi speculated that Dell was worth $12 a share on a sum-of-parts basis, of which the PC business was worth about $4.70. In a report last week, the analyst said Dell could conceivably be split along its PC and enterprises segments, though such an approach would significantly reduce much-needed scale.
DOWNWARD SPIRAL
Shipments of computers by the company, now reinventing itself as a provider of computers and services to corporations and government agencies, plummeted 21 percent in the fourth quarter, according to IDC. In the third quarter, its profit slid 47 percent.
Overall sales of PCs over the holidays slid for the first time in more than five years, according to industry researcher IDC.
On Monday, another industry research firm, Gartner, estimated that Dell lost 2 percentage points of market share in the fourth quarter, slipping to 10.2 percent from 12.2 percent a year earlier.
Dell's fortunes have waxed and waned. Since Michael Dell founded the company in 1984 out of his college dorm room with $1,000, the company has grown into a global PC powerhouse that pioneered just-in-time inventory management and online sales of custom-built computers.
But when Dell handed the reins of his company to long-time lieutenant Kevin Rollins in 2004, sales and customer service began to slip. With the board's blessing, Michael Dell returned in January 2007 to turn his company around, only to run into the global recession and a shift by consumers toward powerful, mobile devices like tablets.
At a Sanford Bernstein investors' conference in 2010, Dell said he had considered taking the company private. He told investors at the time that a transformation of his company that he had hoped to effect upon his return was "incomplete."
Those comments triggered a round of speculation, but most analysts said buying out such a large company would be difficult because of the massive financing requirements.
Michael Dell now owns 244 million shares in the company, according to Thomson Reuters data, and last year was ranked the 22nd richest American with a fortune of $14.6 billion.
Dell's stock soared to an intra-day high of $12.83 in afternoon trade - the highest since May 2012 - after a brief trading suspension. It closed at $12.29.
Its traded bonds also came under pressure over fears of a significant hike in leverage. Its 4.625 percent, 2021 bonds were trading 80 basis points wider at 210 basis points over U.S. Treasuries, while its 2.3 percent, 2015s were about 30 basis points wider at 88 basis points over Treasuries.
"It can be difficult to realize the full value of various corporate assets ... during transition periods, and executing on a long-term transformation as a private company could have advantages," argued ISI analyst Brian Marshall.
(Additional reporting by John Balassi at IFR, Alistair Barr and Noel Randewich in San Francisco, and Jessica Toonkel and Kim Soyoung in New York, Writing by Edwin Chan; Editing by Bernard Orr and Steve Orlofsky)
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