There is a "small chance" lives could be put at risk, the forecaster said.
A yellow weather warning for thunderstorms across the south west of England and South Wales is in place until 6pm on Sunday.
A similar warning has been issued for London, the south east and east of England and the East Midlands until 6am on Monday.
Homes and businesses could be in danger of flooding quickly in the downpours, although the risk is said to be low.
In some parts of southern England, it is predicted that 30-40mm of rain, which amounts to at least half the September average of 55-60mm, could fall.
Buildings could also be damaged by lightning, hail or strong winds.
There is also a "slight chance" of power cuts or that other services to homes and businesses could be lost, while some communities could also be cut off by floodwater, the forecaster said.
People planning on travelling face the prospect of delays or sudden cancellations to trains and buses.
Roads may be closed at short notice due to spray and sudden floods and "difficult driving conditions" are expected on those that remain open.
Heavy rain brought "torrential downpours" across the south west of England on Sunday morning, with localised flooding in south Devon.
The band of rain is expected to move into the south east of England on Sunday afternoon.
Met Office meteorologist Jonathan Vautrey said "there is a chance these thunderstorms turn severe" and bring "gusty winds with quite significant torrential rain".
They will move relatively quickly, making it difficult to pin down where exactly will be worst affected, Mr Vautrey said.
He added: "It is certainly worth keeping up to date with the forecast.
"Although the warning area covers the whole south east of England, not everywhere in that region may see the most severe thunderstorms.
"It is worth checking those things immediately before you head out on your journey so that you are aware where the most severe thunderstorms are possible.
"Make sure you are taking care as the weather could change at very short lead times, and just be prepared for those gusty winds and potentially large hailstorms."
Conditions are expected to remain "blustery at times" early next week but are likely to be fresher.
More storms are possible as the remnants of Hurricane Lee, which hit New England in the US and eastern Canada, is set to move across the UK between Tuesday and Thursday.
It will no longer be a hurricane by the time it reaches UK shores.
Mr Vautrey said: "That will be getting picked up by the jet stream. Showers in places could be heavy with a risk of further thunderstorms.
"It could be quite an unsettled, autumnal week to come."
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