Sky News understands that Liberty has held discussions in recent weeks about an agreed offer for Daisy, which is run by Matt Riley, one of the UK's most prominent entrepreneurs.
The talks are understood to have been discontinued in the last few days because of a gulf in the two sides' views about the potential price that Liberty would pay for Daisy, a source said.
However, Daisy may come under pressure from the Takeover Panel to confirm that it had held discussions if its share price moves significantly during trading on Friday.
The talks about a prospective takeover were held in the wake of a deal struck by the two companies last month. That involved a five-year agreement for Daisy to provide services and engineering support to Virgin Media, which is part of Liberty Global.
It is unclear whether the talks are likely to be revived.
Shares in Daisy closed on Thursday up 1% at 188p, giving the company a market value of just under £500m.
Last year, it paid its maiden dividend, and has used bank facilities to extend the services it offers to small and medium-sized companies through a string of takeover deals.
Liberty has been on a much larger acquisition spree in recent times, snapping up cable and other assets across Europe amid an accelerating consolidation of the industry.
The company, which is chaired by the media tycoon John Malone, has a division called Liberty Global Business Services, into which Daisy would be likely to be integrated if a deal was successfully revived.
Liberty Global and Daisy both declined to comment.