Gusts of up to 50mph are predicted across much of Northern Ireland, with 70mph in some places.
A yellow warning for wind is set to conclude at midnight on Monday. It had initially been due to end at 6am on Tuesday.
Floris is the first storm to hit the UK and Ireland since Storm Eowyn in January, which sparked a rare red warning here.
The Met Office said: “Storm Floris will bring a spell of unusually windy weather for the time of year across the northern half of the UK on Monday.
“The highest winds are expected across Scotland between late morning and early evening. In some areas high winds will be accompanied by heavy rain. Wind will then slowly start to ease from the west later.
“Across the warning area, many inland areas are likely to see gusts of 40 to 50 mph with 60 to 70 mph possible along exposed coasts and high ground, especially Scotland.”
Storm Floris to bring ‘unseasonably strong winds’ to parts of UK
However, as the week progresses the Met Office has forecast temperatures to increase, with Tuesday ranging between 12C and 19C.
On Wednesday, a maximum of 21C is to be expected with a chance of showers on Thursday morning.
Light winds will prevail throughout the week with strong gusts of up to 39mph slowly decreasing as the weekend approaches.
Saturday will see sunny intervals and mild temperatures with a small chance of rain throughout the day.
The Met Office said Storm Floris’ gusts could reach 85mph on exposed coasts or hills in Scotland.
An amber weather warning is in place for the majority of Scotland between 10am and 10pm on Monday.
Train operator LNER has warned passengers not to travel north of Newcastle on Monday.
Services are likely to be impacted in Lancaster, Oxenholme, Penrith, Carlisle, Lockerbie, Motherwell, Haymarket, Glasgow Central and Edinburgh.
Motorists in those regions have been urged to slow down in the poor weather and avoid Highland and coastal routes.
Shaun Jones, of the AA breakdown service, said: “If you’re planning a journey – especially through exposed or rural areas – it’s worth checking the latest forecast, allow extra time, and be prepared for the unexpected.”
In the Republic, Irish forecaster Met Éireann has issued Status Yellow weather warnings, which are due to come into effect from early tomorrow morning.
The forecaster issued a wind warning for counties Clare, Galway, Mayo and Sligo, valid from 2am tomorrow until 1pm.
Another such alert was announced for counties Cavan, Donegal, Monaghan and Leitrim, in place from 4am until 4pm tomorrow.
Storm Floris is the sixth named storm of the 2024-25 naming season, which runs from early September to late August.