He said that some of these wintry showers could drift south to Glasgow and Ayrshire later on Tuesday, with some very low temperatures that night.
The Met Office has issued a yellow warning for snow and ice, running from 9pm on Monday until 3pm on Tuesday.
It says: "A spell of rain, quickly turning to sleet and snow, will move southeast across the United Kingdom on Monday night and Tuesday, gradually weakening across England and Wales".
"A yellow weather warning is in place for snow which may lead to hazardous conditions on roads and footpaths", said gardai.
And forecasters say freezing conditions will bite again tonight with temperatures dropping to -2C (28F).
The Irish meteorological office advised farmers with early-lambing flocks to take special care of their animals as a result of the snow and ice forecast.
Further heavy snowfall is forecast across Scotland, northern England and Wales through much of today (Tuesday).
KentPoliceRoadsThere have been multiple pile ups on the M20
"Total snow accumulation by the end of the week expected to be 20 to 30cm [12 inches] over the hills".
The Met Office Chief Forecaster said: "Further showers of snow, sleet and hail will affect western parts of the United Kingdom, these frequent in western Scotland and the north of Northern Ireland, where thunder is also possible".
Severe weather is possible over the next few days and could affect you. There could then we snow showers in Humberside on Wednesday.
STV weather presenter Sean Batty said: "Most areas will have a wintry spell of weather as a band of sleet and snow crosses the mainland".
"This weather could increase the health risks to vulnerable patients and disrupt the delivery of services". The cold snap is expected to continue until at least the weekend.
Temperatures are also set to plummet to as low as minus 5 degrees in parts of the country.