A yellow - "be aware" - severe weather alert for 2-11am has been issued by the agency.
It said snow could become persistent over the Western Isles and Highlands, with up to 10cm falling.
The warning covers the whole country apart from the east coast, Aberdeenshire, Caithness and the Northern Isles.
A Met Office spokeswoman said: "A band of rain, sleet and snow, accompanied by gusty winds, will move eastwards across Northern Ireland during the early hours of Sunday and then continue into Scotland and parts of northern England.
"For most areas, rain, sleet and snow will tend to only last for two or three hours, but there is potential for sleet and snow to become more persistent over the Outer Hebrides and Highland during Sunday morning.
"Where snow falls, 1 to 3cm are likely, whilst on ground above 200m, 2 to 5cm is likely, and above 400m, accumulations will be higher and close to 5 to 10cm in some places.
"Some travel disruption is possible early on Sunday.
"Icy patches are also likely to develop on untreated surfaces as rain, sleet and snow clears, most likely over Northern Ireland."