Storm Barra is forecast to bring strong winds and heavy rain eastwards into the afternoon before clearing to the north-east in the evening.
The rain may turn to snow, especially over higher ground.
Gusts of 45 to 50mph are forecast, possibly increasing to between 55 and 65mph in the afternoon and early evening along the coast.
The combination of strong winds and snow may cause drifting and poor visibility, particularly on the highest routes.
The Met Office has issued a Yellow Snow Warning for parts of North, West and South Yorkshire. The forecast indicates that snow is most likely in the Harrogate and Leyburn areas.
North Yorkshire County Cllr Don Mackenzie, Executive Member for Access, said of Storm Barra: “As always, our highways teams are ready to respond to any conditions that arise, supported by our 100 farming contractors, should their help be needed in clearing snow on higher ground.
"We also have emergency response crews ready to react to incidents on the roads network.
“While the impact of Storm Barra is not forecast to be as severe as Storm Arwen at the end of November, we would advise anyone planning to travel to consider whether they need to.
"If they do, they should be alert to the conditions and leave extra time for their journey.”
Across the county, residents should secure garden furniture and other loose items that might be susceptible to strong winds.
At the coast, people are advised to avoid seafronts in case of overtopping by waves.
More information about when and where the county council treats highways can be found at www.northyorks.gov.uk/road-gritting
Visit www.northyorks.gov.uk/road-cameras-and-weather-conditions for up-to-date road conditions which can be checked via cameras at key locations around the county.
A winter driving safety checklist can be found at www.northyorks.gov.uk/winter-summer-and-wet-weather-driving-tips