The Met Office has issue a red weather warning for South Wales, meaning there is a likely risk to life.
France was also affected by the storm, especially northwestern Brittany where the Finistere and Morbihan regions were temporarily placed on orange alert for rain and flooding, according to the national weather service, Meteo-France.
Last week, Storm Ciara caused disruption to transport when it hit the United Kingdom.
Britain on Monday grappled with the fallout from Storm Dennis, with several major incidents declared due to flooding and hundreds of warnings still in place after it battered the country over the weekend.
A family is rescued from a property in Nantgarw, Wales yesterday.
The newly-appointed Defra Secretary, George Eustice, denied that the government had been caught off guard by the floods following storms Dennis and Ciara.
The EA said on Sunday afternoon that there been more than 600 flood warnings and flood alerts in place across England.
Met Eireann said that the storm would bring severe and damaging gusts, high waves and a risk of coastal flooding as it continues passing north of Ireland.
Storm Inès and Dennis: Further storms expected from Thursday onwards
Several planes rocked from side to side as they came into land at Birmingham Airport during fierce crosswinds from Storm Dennis. The winds could cause disruption to travel, and there's a chance of some damage to buildings.
A record 594 flood warnings and alerts were raised, extending from Scotland's River Tweed to Cornwall in southwest England.
The Met Business, Britain's meteorological support, only challenges its best warning when it thinks the weather conditions will be so poor there is a "danger to lifestyle".
The agency is predicting that up to 140 mm (about 5 inches) of rain could fall in south Wales before slowly easing later on Sunday.
Gwent Police said that residents of Skenfrith, Monmouthshire, were being advised to evacuate due to the flooding.
Meanwhile, two bodies were pulled from rough seas off the south England coast on Saturday as the storm barrelled in.
ITV forecaster Laura Tobin warned the River Wye is of "most concern" to the Environment Agency as growing water levels continue to rise.
Severe flood warnings were also still in place for the rivers Lymn and Steeping in Lincolnshire, as well as the River Teme in parts of Worcestershire, Herefordshire and Shropshire.
Flights have been grounded for safety reasons, with British Airways and easyJet confirming cancellations.