Met Éireann said that strong to gale force westerly winds will produce gusts of 110km/h (68mph) to 130km/h (80mph) in Connacht and Donegal later.
Elsewhere, Munster, Leinster, Cavan and Monaghan are expected to have 90km/h (56mph) to 110km/h winds.
Met Éireann said it will be extremely windy with some severe, possibly damaging gusts.
“After dark, heavy wintry showers of hail, sleet and snow will develop at first in the north of the country, but during the night they will become more widespread and produce some accumulations of snow, in particular on high ground but also at lower levels,” it said.
“The very strong winds and snow showers will produce some blizzard-like conditions especially on higher ground, with icy patches at all levels.
“The wintry showers will turn to rain showers on Tuesday afternoon as temperatures increase.”
On foot of the prediction, The Road Safety Authority (RSA) has issued an alert to all road users.
#WeatherAlert Met Éireann forecast wintry showers of hail, sleet and snow with possible blizzard conditions and widespread icy patches.
The grandmother of a toddler with Down's syndrome has been waiting a year for a response from the Taoiseach and three government ministers to correspondence about disability cuts referred to them on her behalf by the troika.