A trough of low pressure will extend across New Zealand this week, MetService said, causing the temperature drops.
Unsettled southwesterlies will bring showery weather, falling as snow to low levels for the lower South Island.
The central North Island is also expected to receive snow affecting the higher roads from mid-week.
Monday will start off wet in the west and south of the country, and the south coast of the South Island could even have snow flurries to near sea level on Monday on Tuesday.
Meanwhile, eastern parts of the country will be mostly sheltered in the westerly flow on Monday and remain mainly fine.
Rain could be expected across much of the country during the week, MetService predicted.
"This snow event will not be as widespread or heavy as the last one we had a few weeks ago as the air mass is a little drier," said Ms LaBar.
"However, the mountain passes and high elevation roads could still see some significant snowfall, so be sure to keep up with the latest warnings and road information if you are travelling this week."
Towards the end of the week, the weather should start to improve as southerly winds are replaced by high pressure coming in from the southwest.
The South Island would begin to get drier and warmer by the weekend, Ms LaBar said, but "the North Island is still looking pretty showery".
Mr LaBar predicted temperatures would "gradually rise up" as the week progressed and the southerly winds moved on.
"The North Island will take a little bit longer because the southerlies will be persisting there a littler bit longer, but towards the weekend it looks like things...will start to improve."
She said South Islanders would feel the warm change by the end of the week, with midweek highs in Christchurch predicted to rise from 7C to 10C by the weekend.
Monday forecast:
•Auckland: High of 16C low of 12C. A few showers predicted, becoming more frequent in the afternoon. Westerlies.
•Hamilton: High of 15C low of 4C. A few showers predicted, becoming more frequent in the afternoon. Westerlies.
•Tauranga: High of 16C low of 10C. Fine spells predicted with the odd shower. Westerlies.
•Wellington: High of 12C low of 6C. A few morning showers predicted, then fine. Strong northerlies.
•Christchurch: High of 11C low of -3C. Mainly fine weather predicted with the chance of some showers early or late in the day. Late cold south westerly.
•Dunedin: High of 8C low of 1C. Fine with a few afternoon showers predicted. Southwesterly.
- NZME