Carlyle, which has offered £11bn for Virgin Media, is considering axing the Virgin brand name if its takeover bid is successful, according to a report in the Observer.
The prospect of Virgin rebranding without its link to Branson opens up the possibility for a rapprochement with Rupert Murdoch's BSkyB.
The media giants have been locked in a bitter row since the end of last year, when Murdoch derailed Virgin Media's attempt to take over ITV by acquiring a blocking stake.
Carlyle is understood to have questioned the value of retaining Virgin Media's connection to Branson on the basis that it might be easier to patch up relations with BSkyB without him.
The acquisition of the stake in ITV was then followed by another damaging move by BSkyB to Virgin Media.
In February, the launch of Virgin Media's TV offering was troubled by problems after BSkyB withdrew channels such as Sky One and Sky News from the Virgin Media offering.
The disappearance of the channels has increased churn at Virgin Media, as customers cancelled their subscriptions because the cable company could no longer offer first runs of hit shows such as '24' and 'Lost', which run on Sky One.
However, Carlyle is believed not to have made its mind up yet about dropping the Virgin name, as the plans would involve overcoming legal issues, such as an existing licensing agreement with Virgin.
Virgin Media is not owned by Branson, instead he owns 10% of the shares and rights over the Virgin name. He is also one of Virgin Media's largest shareholders through his company, Virgin Entertainment.
Last year, Branson sold his Virgin Mobile company to what was then NTL, before the cable company rebranded as Virgin Media.
The sale allowed Virgin Media to become the UK's first quad-play media company: offering customers TV, mobile, broadband and a landline in one package.
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