The Met Office have released an amber "be prepared" warning for the area.
It said: "Snow showers will be frequent and heavy at times, with drifting and temporary blizzard conditions in the westerly gales.
"Amounts of snow will be variable, but accumulations of 4-8in are expected in some areas, especially on routes above 300ft.
"The public should be prepared for possible disruption, especially to travel."
John Lee, a forecaster with MeteoGroup, said: "It's going to be very windy through Today and into tomorrow, with near gale or gale force winds from the North West bringing wintry showers.
"The showers will turn to rain today, but there will still be a risk of some snow.
"Tomorrow, the strong winds will continue, but it should be drier, although there will be a risk of ice."
The West was hit by gusts of up to 70mph and many ferry services were disrupted, with cancellations in the Western Isles, including the Oban-Lochboisdale sailing.
The strong winds forced the closure of the Erskine Bridge, Clackmannanshire Bridge, Skye Bridge and Forth Road Bridge to high vehicles.
Scotland's Transport Minister Keith Brown urged people to prepare for more potential transport disruption today.
Mr Brown said: "Heavy snow, combined with gale force winds, will mean a testing journey for many this evening and during rush hour tomorrow.
"We would expect to see difficult conditions emerging on our roads, as well as the potential for disruption to all transport.
"We are delivering winter service treatments to our roads, but sudden weather changes can happen and the impact the weather can have from one location to the next can be very varied.
"People should stay up to date with live travel information. They can access the Traffic Scotland website, internet radio bulletins and Variable Message Signs that are being used to provide up to the minute information as the situation develops."