Storm Inès will arrive on Thursday, followed by Storm Dennis over the weekend.
After last weekend's storm, river levels are high; the ground is saturated and trackside trees have been weakened, this is why operators are advising passengers to check before travelling.
Widespread damage from flooding and disruption to transport is expected with the worst hit areas possibly experiencing gusts of 80mph and up to 140mm of rainfall.
Last weekend teams of engineers worked to remove fallen trees and other debris from the railway.
Meteorologist Matthew Box said: "The winds will be a touch down on what we saw with Ciara but the focus is on a bit more rain".
The Met Office has heightened weather warnings across the United Kingdom due to the risk of flooding posed by Storm Dennis.
The Met Office issued the amber warning this morning, with heavy and prolonged rain expected from just after midnight on Sunday right through until 6pm.
While Storm Dennis is not predicted to be as severe as Ciara, wet and windy conditions are expected to continue until Tuesday.
Storm Dennis' arrival on Sunday has prompted another yellow warning until noon on Monday, for winds gusting to 70mph, affecting the whole of Scotland apart from Orkney and Shetland.
Last weekend, Storm Ciara brought gusts of more than 90mph in places.
Motoring groups have urged drivers to take extra care on the road and prepare for journeys to take longer because of the weather.
Caroline Douglass, flood duty manager at the Environment Agency, said: "Remember to never drive or walk through floodwater".
An EA spokesman said: "Storm Dennis will bring widespread, heavy and persistent rain over the weekend".
There has been localised flooding in various parts of County Derry this week.
"River and surface water flooding is also possible widely on Sunday across parts of the south and south-east of England and is most likely in southern parts of that area".