In a statement, the agency said: "We're warning communities across south west England and southern Wales to be vigilant and alert for flooding as heavy rain is set to return on Sunday afternoon and into Monday." It added that Sunday's expected deluges may result in significant disruption.
Gemma Plumb, a forecaster for MeteoGroup, the weather division of the Press Association, said the rain should have cleared all parts of the UK by late morning.
"Once the rain in the south and east of England has cleared most of the UK will be dry with sunny spells and temperatures between 13C and 16C," she added.
"Tonight looks like most places will be dry, but it will be a chilly night with mist and fog patches in places. Temperatures could touch zero in some areas of the north with the potential of a local frost tomorrow morning."
Ms Plumb said Sunday will get off to another dry and bright start with long spells of sun before the next band of heavy rain moves in later on in the day. "The rain will move into south west England, Wales and central-southern England and spread north-east. It could be quite heavy and persistent and will continue into Monday."