A yellow ‘be aware’ alert is valid from 8am on Wednesday until 9am on Thursday, covering the city and large parts of England and Wales.
The large warning area is due to the complexity of the forecast, which means it is not yet certain which areas will be most badly hit.
Strong and gusty winds are also expected, with gales possible in exposed areas.
The Met Office is warning some localised flooding may occur with a chance of travel disruption.
A spokesperson for the Met Office said: “A complex area of low pressure is expected to deepen over the Bay of Biscay later today and into Wednesday, before pushing north towards the UK.
“Rain will arrive into southern areas Wednesday morning, before spreading to the rest of the warning area through Wednesday afternoon and into Thursday.
“Rain should ease from the south during the day, though some heavy and thundery showers may develop across the southeast and East Anglia through the afternoon and evening.
“Most parts of the warning area should see 15-30 mm of rain, however some places could see as much as 40-70 mm.”