Met Eireann, which named the storm, has issued a nationwide orange wind warning for Monday.
Forecasters are predicting that winds will reach speeds of up to 54mph in Portsmouth and will be strongest in the evening.
It adds: "Cork County Council expects the main impact from Storm Brendan to include fallen trees, possible structural damage from high winds and coastal flooding caused by the storm surge". However, at 3pm on Sunday the forecaster extended the warning to the entire 26 counties.
The National Directorate for Fire and Emergency Management (NDFEM) issued a statement on the situation on Sunday.
"Members of the public are advised to stay high, stay dry and to stay away from the coast, rivers and lakes".
A calm start to the morning is expected in Dublin, although conditions will worsen as the afternoon approaches and the remnants of Storm Brendan take their toll.
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The directorate said local authorities had activated coordination and crisis management arrangements to prepare for the arrival of Brendan.
A warning for Connacht, Donegal and Kerry is in effect from 5am on Monday until 9pm that night.
It is also warning there is a significant risk of coastal flooding.
According to details on the Met Office website tomorrow will be "a widely very windy period" with "some disruption to travel likely".
The Met Office has issued a yellow wind warning for Monday afternoon and evening, this covering Northern Ireland and much of the western half of the United Kingdom, as well as northeast Scotland.