Rupert Murdoch's News Corporation is seeking to take full control of satellite broadcaster BSkyB, by acquiring the 60.9% of the shares it does not already own.
News Corporation's initial offer of 700 pence a share, which values BSkyB at around £12bn, was rejected.
Talks are now expected to continue until a deal is reached. BSkyB said it wants "in excess of 800p" per share.
News of the bid pushed shares in BSkyB up 16.6%.
News Corporation also owns the Sun, Times and News of the World newspapers.
Agreement between News Corporation and BSkyB would go some way to clear the way for Mr Murdoch's ambitions, though any deal would also need regulatory approval.
"Both parties have agreed to work together to proceed with the regulatory process in order to facilitate a proposed transaction," said News Corporation.
BBC business editor Robert Peston said that if the deal did go ahead, it would "erase any scintilla of doubt that Mr Murdoch's News Corporation would be the most powerful of all the traditional media groups in the UK".
"The combination of Sky with his newspapers, such as the Sun and the Sunday Times, would generate annual revenues of around £8bn, compared with the £4.6bn income of the next largest player, the BBC."
Our editor added that any agreed deal between News Corporation and BSkyB may cause problems for the UK's coalition government.
He said this was because while the Conservatives had benefited from the support of News Corporation's newspapers during the general election, the Liberal Democrats were far more hostile to Mr Murdoch's media empire.
Whether to grant regulatory approval for any agreed takeover would be decided by the European Commission - unless it was called in by either Business Secretary Vince Cable or the Office of Fair Trading (OFT).
Mr Cable would have to say he had public interest concerns, while the OFT would have to stress competition fears.
British Sky Broadcasting was formed in 1990 by the merger of News Corporation's Sky Television and rival British Satellite Broadcasting.