The agency issued a yellow - "be aware" - severe weather warning for western and central Scotland, Wales and the west coast of England.
It said there was a "small chance" of homes being flooded and fast-flowing or deep floodwater posing a danger to life.
Flooded roads could also cut off isolated communities and fallen trees causing power cuts.
Up to 100mm (4in) of rain is expected on higher ground, with winds gusting to 60mph.
The warning will be in force for the whole of Sunday until 6pm on Monday.
It covers the western half of the Central Belt, south-west Scotland and as far north as Pitlochry.
However, uncertainty over the impact mean the Met Office has not yet classed the bad weather as Storm Brian - which would be the second of the autumn and the first to hit Scotland.
A Met Office spokeswoman said: "Spells of heavy and persistent rain are expected over some areas of western and northern Britain, accompanied by strong winds at times.
"There is a small chance homes and businesses could be flooded, causing damage to some buildings and some communities could become cut off by flooded roads.
"Where flooding occurs, there is a slight chance of delays to train and bus services, whilst spray and flooding could lead to difficult driving conditions and some road closures.
"There is a small chance of fast-flowing water or deep floodwater causing danger to life.
"In addition, strong winds may be an additional hazard at times, with a slight chance of trees falling, power cuts and loss of other services to homes and businesses.
"Many places, particularly high ground, will see 25-50mm of rain, but where the rain becomes prolonged as much as 100mm could fall over upland areas.
"The exact location of where the largest rainfall totals will occur is currently uncertain.
"Although it looks as if the more persistent rain should clear away southeastwards later on Sunday, further heavy persistent rain is possible in places on Monday.
"This appears more likely across the more south-western parts of Britain.
"Strong winds are likely at times, with gusts of 50-60 mph possible in some exposed coastal areas."