Rain will then hit Edinburgh on Friday afternoon, becoming increasingly heavier into the evening.
The yellow warning has been issued for strong gusts of wind up to 60mph - and up to 70mph in exposed and hilly areas - which would be coupled with very heavy rain.
While the strongest winds will have passed by Friday noon, Ms Dillon said it would continue to be windy right through Friday and into Saturday and be accompanied by heavy rain.
It comes as Met Eireann has issued an orange weather warning for coastal counties including Galway from 10pm tomorrow until Friday morning.
Strong winds associated with Storm Callum, coinciding with high spring tides, will affect Ireland on Thursday night and Friday morning.
Storm Callum is expected to bring gale force winds of up to 150 kmph with torrential rain.
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Winds will be strongest overnight and early on Friday, gusting between 110kph and 130kph at coasts.
The extreme conditions triggered two yellow - "be aware" - warnings from the Met Office after a similar alert on Tuesday caused flooding and travel disruption to roads, railways and ferries.
The Met Office said that "leaves and twigs, brought down by the wind, could block drains and culverts, increasing the likelihood of flooding".
"The public are asked to be careful on all roads due to surface water flooding and in coastal areas, especially at high tide".
The agency said Saturday's warning for rain could see more than 120mm falling.
"It's not a major event in the east, we don't expect a major flooding event along the east coast", she told RTÉ Radio One's Today with Sean O'Rourke programme.
Conor Faughnan, AA Director of Consumer Affairs said: "It's vital that we all take a look at our own road behaviour and adapt it accordingly to minimise incidents during this weather". There is also the risk of further severe Southerly winds impinging onto Southern and Eastern coastal areas during the day.