Met Éireann has issued a status yellow warning with the possibility of snow on higher ground in counties Donegal, Leitrim, Mayo and Sligo.
The forecaster has warned of heavy showers and hail on Monday morning, particularly in Connacht, which will turn increasingly wintery as temperatures drop.
However, most of the country will become dry with good sunny spells as the afternoon develops.
It will be a cold day across the State with temperatures of 5 to 9 degrees.
Temperatures are expected to drop to freezing this evening with frost leading to a risk of icy patches on roads.
Any frost and ice will clear overnight as milder, windy conditions spread from the west.
Tuesday is expected to be mild and windy with temperatures of 10 to 12 degrees and scattered outbreaks of rain throughout the day.