The coldest morning so far this year gripped Melbourne on Sunday, when temperatures fell lower in the city than inside the average Australian fridge, and the bureau is warning of more icy conditions and bitter cold in the week ahead.

Temperatures in the CBD fell as low as 3.3 and 6.6 degrees on Sunday and Monday mornings, and to below-zero temperatures in the city's north-east.

The Bureau of Meteorology is forecasting a top temperature for Melbourne of just 12 degrees on Monday with a very high chance of showers throughout the day.

And there is more cold to come as the state enters what senior forecaster Michael Efron has described as the state's "heart of winter".