Internet service provider Breathe has yet to reveal whether it will reimburse an estimated 50,000 subscribers to its unmetered access service, which ceased on Sunday after being deemed "no longer viable".
The firm wouldn't comment to the BBC.
An insider said helplines set up to deal with customer queries "have been busy" but declined to say anything further.
Breathe, which launched the Breathe Freely service in April, faces a bill of about £2m if it agrees to reimburse subscribers.
The firm is now looking for a buyer to save it from receivership, the Sunday Telegraph said.
Series of setbacks
The closure is the latest of a series of setbacks to hit firms attempting to offer unmetered access to the internet - allowing unlimited surfing for a one-off fee.
In July, internet service provider (ISP) Line One withdrew a flat rate package, after finding that a small number of surfers, by logging on 24 hours a day, rendered the service unprofitable.
The same month Breathe itself withdrew unmetered access connections from 500 heavy users, who were offered a refund.
And AltaVista in August admitted that, because of cost concerns, it had never launched its high profile flat-rate service, despite signing up 270,000 customers.
Industry shake-up
Firms attempting to offer unmetered access have suffered through the fact that while they offer a flat rate connection, they are charged by the minute by the telecoms companies which own the lines and exchanges through which the service operates.
British Telecom is currently rolling out an unmetered access package, called Friaco, which will allow ISPs also to benefit from flat-rate infrastructure charges.
"Developments in the wider unmetered access market indicate that the only sustainable model for such services is based around Friaco," Breathe said in a statement on Friday.
AOL last month became the first ISP to take advantage of BT's Friaco package.
New economy entrepreneur
Breathe is controlled by Martin Dawes, the man behind mobile phone firm Martin Dawes Telecommunications, which BT Cellnet bought in March for £130m.
A syndicate in February paid £20m for a 35% stake in Breathe, which has also been backed by venture capital giant 3i, the Sunday Telegraph said.
Only three months ago Breathe was valued by analysts at £100m, the newspaper said.