Network Rail said fallen trees and other debris can blow onto the line, with the potential to damage train-powering electric wiring or block the track.
On Thursday, the Met Office issued a rare red weather warning for some parts of the UK, which means trains will need to run at lower speeds to be safe, resulting in a significant reduction in services.
In Sheffield, a yellow weather warning has been issued for the city, with the Met Office forecasting gusts of up to 60mph this afternoon.
The rail service said trained workers have been deployed to key locations across the region to respond to issues quickly, as strong winds and heavy rain are expected to cause widespread disruption for the railway.
Matt Rice, North & East Route Director for Network Rail, said: “Running a railway through extreme weather is always a challenge.
"We have extra workers out on the network at key locations, ready to react quickly to Storm Eunice and repair the railway wherever it’s safe to do so.
"As the weather worsens I’d advise passengers to avoid travelling on Friday and, wherever possible, try to replan your journey when conditions improve.”
National Highways has also warned motorists to expect severe driving conditions across much of England and a significant risk to vehicles in the South West, where a red alert for gales has been issued in addition to a Met Office Red National Severe Weather Warning.