Isabel dos Santos has offered €1.35 a share for Portugal Telecom, which is controlled by Brazilian telecommunications group Oi.
The company described the bid as inopportune, ill-timed and inappropriate.
Her move is an attempt to derail a €7bn offer for Oi's Portuguese assets by French company Altice.
Oi is still considering the offer made last week by Altice, which is controlled by Franco-Israeli billionaire Patrick Drahi.
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Altice already owns two small cable companies in Portugal and buying the assets of the former state-owned monopoly would help it compete with rivals Vodafone and Optimus.
The offer from Ms dos Santos's Terra Peregrin represented an 11% premium to the closing price of Portugal Telecom on Friday and sent the shares up by 12% to €1.37 on Monday.
Oi is considering selling the Portuguese assets it acquired by taking over Portugal Telecom to pay down some of its $18bn debt mountain.
The Brazilian company's merger with Portugal Telecom soured in the summer after the Rioforte holding company of the Espirito Santo banking family - ultimately a shareholder in PT -  defaulted on nearly €900m of debt owed to PT. Oi said it had not been aware of the debt before the merger.
As a result Portugal Telecom's stake in Oi was slashed, leaving it with just 25.6% of Oi rather than the 38% originally agreed. It also led to the resignation last month of Zeinal Bava, the chief executive of the new company and former chief executive of Portugal Telecom.
The 25.6% stake in Oi is the only asset now held by Portugal Telecom.
Shares in Portugal Telecom have fallen by about 60% in recent months following the debt scandal.
Shares in Portugal Telecom have jumped following a €1.2bn (£940m) bid by the Angolan president's daughter.