SYP Operational Support said they are already getting reports of heavy snow and strong winds on higher ground and vehicles becoming stuck.
In a statement, they said: "If you do have to drive then make extra time for your journey. Plan ahead now.
"You can check your proposed routes for any closures on most mapping apps or Highways Englands app or website.
"Check the weather forecast. Ensure your vehicle is fit for purpose, paying particular attention to your tyres.
"Carry extra warm clothing in case you get stranded. An important message to get across though is to drive to the conditions and be patient.
"Steady on the pedals, and the steering wheel. No journey is worth the risk to yours and others safety."
A yellow warning for snow and ice has been issued in South Yorkshire on Sunday (February 7) as Storm Darcy is set to barrel through the UK in the next few days.
The Met Office said the weather warning for the county is valid until Wednesday (February 10) while some other parts in the UK will face significant snow accumulations with amber warnings in place.
This happens when cold air emanating from Russia and Eastern Europe will move across the UK over the coming days, bringing significant snow accumulations to parts of Eastern England and Scotland.
Rain will then increasingly turn to snow on Sunday as the cold air spills across the whole of the UK.