Forecasters warned that more than two inches of rain could fall within 12 hours on Monday in some parts of the country.
The Met Office has issued a yellow weather warning covering large swathes of England between 12 midday and midnight.
The entirety of the Black Country, Birmingham and Staffordshire are included in the warning, which stretches from Devon on the south coast to South Yorkshire.
The warning came as drivers continued to get stuck in floodwater in Albrighton throughout the weekend despite "road closed" signs being in place.
The main road into Albrighton, off the A41, passes under a railway bridge which has been flooded for weeks and could get worse as more rain arrives.
The Met Office said the wet weather will move northwards across much of England on Monday, with the heaviest rain forecast for the afternoon and evening.
While up to 25mm is likely to fall “quite widely”, some areas could see up to 60mm (2.3 inches), the Met Office said.
“This brings a risk of flooding and disruption, but the situation is uncertain, such that the likelihood of this across any given county is small,” the Met Office said.
Flood alerts are in place for the River Severn from Shrewsbury to just before Bewdley, including Coalbrookdale, Ironbridge and Bridgnorth, and for the Severn Vyrnwy confluence from the Welsh border at Llawnt to Shrawardine near Shrewsbury.
However Bewdley, Stourport-on-Severn and Worcester are not yet included in the flood alerts.
Other locations that may be affected include Llanymynech, Maesbrook and Melverley, according to the alerts which warned drivers to avoid low lying roads near rivers.
Most of the Black Country and Staffordshire should be dry until around midday on Monday when heavy rain is likely to set in until at least 10pm, according to the Met Office.
Met Office meteorologist Mark Wilson said there is an “improving picture” for Tuesday, with temperatures reaching up to 16C.
He said while there will be some rain in Northern Ireland on Tuesday, it will be “drier” for the rest of the UK.
The wet weather will follow an “unsettled” Sunday across England, Wales and Northern Ireland, Mr Wilson said, with the Great Birmingham Run course having to be cut short as a result of the weather.
Heavy rain was forecast for the Midlands, northern England and Northern Ireland, with some areas seeing up to 20mm fall throughout Sunday.
ℹ️ | For safety reasons we have been required to reduce the length of the course today.— Great Run (@Great_Run) October 13, 2019
We apologise to runners for the late change but the safety of all participants is our number one priority.
We’ll provide a further update later.
Meanwhile, gusts of around 20mph to 30mph are forecast for East Anglia and the North East coast.
Mr Wilson said: “It is looking pretty unsettled.
“Heavy rain across England, Wales and Northern Ireland will push northwards throughout the day.
“We have not got any weather warnings out [for Sunday], but the rain could cause [road] surface spray, so that could lead to some difficult driving conditions.”
Mr Wilson said the wet weather will push north throughout the day into southern Scotland.
It will be followed by drier weather moving up from the South and South West from the mid-afternoon, Mr Wilson said.
South-east England is forecast to see the warmest temperatures at 18C, while the north of England will see the coolest temperatures – between 11C and 12C.
Mr Wilson said the rain will “ease” overnight on Sunday and will be followed by fog patches across England, Wales and Northern Ireland.