The announcement yesterday followed disruption to commuter services as drivers refused to work on rest days as part of an industrial dispute.
Connex had already published the times it would be transporting New Year revellers to and from London on trains driven by workers being paid a special pounds 1,000 bonus for volunteering to work on 31 December and 1 January.
A spokesman for Connex said: "If drivers follow the union's call not to work on a voluntary overtime basis, then clearly we are not in a position to guarantee those services. We cannot advertise services and then strand people, with the possibility that we can't get them home.
"We would therefore have to withdraw those services. If we do, then each driver stands to lose pounds 1,000.
"We very much regret the inconvenience and disappointment this will cause customers. Obviously we'll try to get both sides moving towards a resolution as we'd be very sorry to have to carry out this action."
Yesterday a number of trains were affected by long delays and cancellations in network areas covered by the Connex franchise, including Kent, Surrey and Sussex
Services from Horsham in West Sussex and Redhill, Surrey, suffered particularly badly from disruption.
The position was made worse by train cancellations and points failures, which were blamed on the cold weather.
The company, which employs about 380 drivers, could not say how many of its drivers had to work on rest days.
A spokesman said the company was reducing its dependence on rest day working, but conceded it had not set a target to employ enough drivers to eradicate the practice.
The drivers' union, Aslef, told Connex that the action was being taken because of the lack of progress in talks aimed at cutting drivers' hours from 39 to 35 a week, and making improvements to pensions. A spokesman said the union had a separate agreement with Connex for working over the millennium weekend, which should not be affected by the dispute over hours.
A Connex spokesman said the company was "mystified" by the action, because it believed the talks had been making progress.
"All workers rely to a certain degree on doing overtime," a Connex spokesman said.
"Aslef has withdrawn this because they say Connex has failed to enter into 'meaningful' talks."
Connex had announced that it planned to run trains up until 10pm on Friday, 31 December and then to resume services from 1am on New Year's Day until 4am.Reuse content