Mr Murdoch was reportedly offering up to $160m for the stake in the loss- making Stream, which has 80,000 subscribers.
The success of the negotiations, sources said, hinged on approval of the deal from the state broadcaster RAI.
Earlier this year RAI reached an outline accord for cooperation with Stream in the digital television sector.
Mr Murdoch's negotiations with Telecom Italia entered their final stretch amid persistent rumours that the News Corporation chief will also lodge a bid for pay-television rights to all Italian Serie A and B soccer league matches.
According to local press reports, Mr Murdoch is prepared to put up L4.2bn (pounds 1.4bn) to secure these rights for six seasons.
At a meeting of the Italian Football League yesterday League Chairman Franco Carraro said: "No written or verbal offers from Mr Murdoch were received.
"However, the information I have to hand leads me to believe that things are moving fast."
Mr Carraro said any forthcoming bids to compete with Telepiu - which has reached a pounds 700m deal to transmit home matches played by Juventus, AC Milan, Inter and Napoli from 1999 to 2005 - would be welcome.
Telepiu is currently Italy's only broadcaster of encrypted soccer games, with 360,000 subscribers to its digital channels.
Mr Murdoch has been seeking to enter the flourishing Italian television market for more than three years.
As recently as 2 September Mr Murdoch was in Italy to discuss a pan- European bail-out plan for the ailing German media group, Kirch, with Silvio Berlusconi, the former prime minister and the majority shareholder of Italy's largest private broadcaster Mediaset SpA.
As with three earlier rounds of talks with Mr Berlusconi, those contacts came to nothing.
But with his efforts looking increasingly likely to be crowned with success yesterday, the hostility that Mr Murdoch has always aroused in Italy came to the fore.
Telecom Italia shares plunged 5.93 percent on the Milan Stock Exchange, as Telecommunications undersecretary Vincenzo Vita warned that "once he has entered the digital television field, Mr Murdoch will certainly not limit his interests to football.
"I don't want to demonise the man, but he really worries me."