Members of the drivers union Aslef are due to walk out for three days in the dispute over driver-only trains.
Southern's parent company Govia Thameslink (GTR) has asked the Court of Appeal to overturn last week's High Court refusal to ban the strike action.
A decision will be given at the end of the hearing, due to last all day.
Lord Justice Elias, sitting with Lord Justice Lewison and Lord Justice Lloyd Jones, have been asked to overturn the refusal to grant an injunction blocking the "unprecedented" action.
GTR has already been hit by a series of strikes by the RMT union in a long-running dispute over the role of conductors.
Drivers are due to strike on Tuesday, Wednesday and Friday, with a week-long walkout planned from 9 to 14 January.
Southern said given the timing of the Court of Appeal's decision, it would be too late to get trains and drivers in the correct position for Tuesday, and any services it could run would be "extremely limited".
Angie Doll, passenger services director, said: "We hope for the sake of our passengers our application to the Court of Appeal is successful.
"Even if we are able to stop the strikes through the court, services will still be very heavily  impacted tomorrow.
"We will work through the night to try and provide as many services as possible, but we are still advising passengers not to travel as we will not be able to offer a robust service they can rely on.
"We are sorry but the industrial action by Aslef leaves us no viable alternative, but be assured we will do all we can overnight," she said.
The RMT union, which has been staging strikes since April, is also planning further stoppages either side of Christmas.
00:01: Tuesday 13 December to 23:59 Wednesday 14 December (Aslef)
00:01 Friday 16 December to 23:59: Friday 16 December (Aslef)
00:01 Monday 19 December to 23:59 Tuesday 20 December (RMT)
00:01 Saturday 31 December to 23:59 Monday 2 January (RMT)
00:01 Monday 9 January to 23:59: Saturday 14 January (Aslef)
Strike-hit rail operator Southern has warned passengers not to travel on Tuesday, irrespective of the outcome of a last-ditch bid to halt a walkout.