The Met Office said between five and 10cm could fall over high ground as temperatures plummet across the region.
Counties Armagh, Antrim, Down, Londonderry and Tyrone could be affected.
The public have been warned to expect some travel disruption.
"During Tuesday an area of sleet and snow over Scotland will move south into parts of Northern England and Northern Ireland," the Met Office said. "Ice will also be a hazard."
Police have also warned motorists of black ice in the Tornaroy Road area of Belfast.
It follows a number of traffic collisions on Sunday.
The Roads Service said it has salted the main network overnight.
"Salting of those roads on the scheduled salting network considered to be at risk from frost or ice was completed last night," a statement said.
"Further treatment is underway this morning where required. Drivers are advised to exercise caution especially when travelling on untreated roads."