The yellow - "be aware" - alert for up to 40mm of rain covers the east coast north from Fife to Aberdeenshire.
It will be in force from midnight to 11pm.
Snow could also fall for a short time above 400m.
A spokeswoman for the agency said: "Rain will turn persistent and occasionally heavy across parts of eastern Scotland, bringing the risk of localised flooding on Saturday.
"An active frontal system will push up from the south on Friday night and Saturday morning, only slowly clearing north later on Saturday.
"20-30mm of rain is expected to fall quite widely, while some areas could see around 40mm, mainly over higher ground.
"A short period of snow is possible on higher ground above 300-400m before it turns to rain."
The Met Office issues such colour-coded severe weather warnings when there is the potential for widespread disruption or to danger to life.