Germany's largest bank plans to fund the acquisition with the 10.2bn-euro (£8.9bn, $14bn) capital increase it successfully completed on Wednesday.
Deutsche Bank currently owns a 30% stake in Postbank and wants to take full control by buying the remaining shares.
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Postbank's shareholders have until 4 November to sell at 25 euros a share.
Deutsche Bank said the takeover was aimed at "expanding its strong domestic position", particularly in retail operations.
On Wednesday, the bank announced that it had successfully increased its capital by selling 308.6 million new shares at 33 euros a share, in the biggest such move in its history.
This raised its market capitalisation from 25.7bn euros to more than 38bn euros at current share prices, the bank said in a statement.
Deutsche Bank had been intending to take control of Postbank in 2008, but the move was delayed by the global financial crisis.
Deutsche Bank agreed in 2009 to buy a further 27.4% in 2012 from Postbank's main shareholder, Deutsche Post, for 45 euros per share.
Many of Postbank's minority shareholders were disappointed by the 25 euro-per-share offer price, the legally required minimum.
According to German media reports, an association of small shareholders is planning to take legal action in an effort to obtain a better offer.
Deutsche Bank is Germany's largest bank, but has a greater presence in investment banking than in the retail market. Postbank is Germany's largest retail bank, with 24 million clients.