Will Eddie Lampert take Sears Holdings (NASDAQ:SHLD) private? Will he sell off its vast real estate holdings that house Sears and Kmart stores all over the U.S.? Or will he turn this once-mighty retailer into a Warren Buffett-style portfolio of companies, a la Berkshire Hathaway (NYSE:BRK.A)?

It seems like every day lately, a new rumor is running through Wall Street about what Lampert is up to at the struggling retailer.

Whatever plans the hard-to-pin-down hedge fund manager is cooking up, Sears shares sure have been on a roll, rising some 69% in the previous two trading weeks, from opening at $29.20 on Jan. 9 to closing at $49.48 on Jan. 20. They’re trading another 9% higher in morning trading today.

One group keenly interested in the stock’s latest jump must be the well-heeled investors in Lampert’s hedge fund, ESL Investments. Among the largest of those shareholders (in addition to Lampert himself) is Goldman Sachs (NYSE:GS) and several of the giant investment bank’s clients. According to The Wall Street Journal, “Clients of Goldman invested about $3.5 billion in Mr. Lampert’s hedge fund through a special deal more than four years ago. Goldman invested about $75 million of its own money as part of the arrangement.”

The Journal points out that this money is locked up in ESL Investments through the end of 2012, and through last Friday the Goldman clients’ stake was up just 1% overall. With Sears retrenching in its retail efforts, closing some 120 more stores nationwide, its rivals — including Target (NYSE:TGT), Kohl’s (NYSE:KSS) and Wal-Mart (NYSE:WMT) — are circling with the aim of picking off ever more of Sears’s customers.

No wonder the Street is rife with rumors about Lampert’s next move will be.