The German Financial Markets Stabilisation Fund (SoFFin) said it would offer €1.39 ($1.82) per share, around 10 percent more than the minimum the government could offer (€1.26.)
Shareholders have until May 4 to accept the offer, the statement said, adding the government intended to take "full control" of the bank - which is likely to cost Germany €290 million.
"In view of the serious situation in which Hypo Real Estate... currently finds itself, we consider €1.39 to be an attractive price," said Hannes Rehm, chairman of the SoFFin Management Committee.
He added: "It is in everyone's interest, the financial markets', the company's, its employees' and customers', to get this done promptly."
Berlin is concerned that a collapse of HRE would prompt the sort of financial market chaos that followed the failure of US investment bank Lehman Brothers in September.
he government passed an emergency bank nationalisation law giving it the power to seize investors' share by force if necessary. The bank's nationalisation could come at the expense of US investor Christopher Flowers who heads a consortium owning almost 24 percent of the shares in HRE.
Flowers has played his cards close to his chest, saying he will study the offer, but would prefer to remain a shareholder.