National Highways Yorkshire confirmed that snow and sleet is expected across higher routes in the county this evening.
All the roads are being treated overnight but "salt and grit alone won't stop snow settling", National Highways update said.
It continued: "If heading out later, please drive to the conditions and be prepared for adverse wintry weather."
Britons have been warned to expect freezing conditions in the coming days, as the record-breaking mild weather over Christmas and the new year is replaced by sub-zero temperatures.
A cold weather alert is in place for most of northern England, with severe icy conditions and potentially heavy snow from midnight on Tuesday into lunchtime on Thursday.
It comes as warm winds from the mid-Atlantic meant temperatures hovered around the 16C mark in parts of the UK at the turn of the year, making it the warmest New Year’s Eve and New Year’s Day on record.
Met Office chief meteorologist Neil Armstrong said: “Following an exceptionally mild spell, temperatures are set to drop for many in the north of England from late on Monday and into Tuesday.
“This drop in temperatures will spread further south later on Tuesday.
“This change is being brought on by a cold northerly flow, which brings with it the risk of some wintry showers in places in northern England, especially over high ground.
“Minimum temperatures over the next few days are expected to drop below freezing, heightening the risk of some ice formation.”
The UK Health Security Agency urged people to check on vulnerable members of the public in the coming days, including the elderly and those with serious illness.
UKHSA consultant Agostinho Sousa said: “Cold weather can have a serious impact on health, particularly for older people and those with heart and lung problems, as it increases the risks of heart attacks, strokes and chest infections.”