A week after a bankruptcy judge approved its wind-down plan, the estate of Lehman Brothers Holdings Inc. is coming out swinging.

Facing a fight for control over its largest real-estate asset—apartment company Archstone—the estate of the failed investment bank is planning to make a $1.33 billion cash offer for a chunk of Archstone it doesn’t already own, said people familiar with the matter.

The move, which is subject to bankruptcy-court approval, would pit Lehman against Sam Zell’s Equity Residential, which earlier this month agreed to acquire a 26.5% stake in Archstone from Bank of America Corp. and Barclays PLC. Mr. Zell has said he is trying to take over Archstone, which owns stakes in more than 70,000 apartments concentrated in major U.S. cities.

The battle is shaping up at a time when the market for rental apartment buildings is hot. Rents and occupancy rates are rising because of increasing demand from the millions of would-be homeowners getting closed out of that market by foreclosure and tight credit conditions. Investors have bid up the price of apartment buildings in the hottest markets to record levels.

sam zell photo

Lehman’s move to match Mr. Zell’s bid would be a risky one for an estate that is generally looking to sell assets, not add them. The Equity Residential bid values the company at about $16 billion, including $11 billion in debt. That translates into an initial yield of 5.3%, according to Equity Residential, a relatively low level comparable to the yields seen in properties sold in 2005 and 2006 as the market neared its peak.

Lehman became involved in Archstone during the boom years, leading a $22 billion leveraged buyout of the apartment giant in a partnership with Bank of America and Barclays. When the dust from Lehman’s collapse settled, Lehman owned a 47% stake in Archstone, while the other banks owned 53%.