Forecasters at the Met Office warn that this could cause dangerous flooding, and damage to homes and businesses is likely.
Three new amber warnings covering the highlands and associated rivers were issued for a 24-hour period over the weekend.
Forecasters predict that "very heavy rain" will hit the areas around the Brecon Beacons in South Wales, the Yorkshire Dales and Dartmoor in Devon from 3:00 p.m. Saturday to 3:00 p.m. Sunday.
On Sunday, a fourth amber warning for areas south of London, stretching from Andover in Hampshire to the east, was also issued shortly after midnight until 6 p.m.
Rapid or deep floodwaters are also likely to be "life threatening", while trains, buses and roads could be affected by closings, cancellations and delays.
There is a risk that communities will be cut by flooded roads and power cuts in the affected areas.
Three large yellow warnings for strong winds and heavy rain also cover the whole of England, Wales and the south of Scotland between 9 am and noon Saturday.
Rain and wind are expected to head east throughout the weekend, with warnings for England and Wales continuing until late Sunday evening.
The wind warning over Scotland will increase Sunday to encompass the entire region, including Northern Ireland, persisting until noon Monday.
The arrival of Storm Dennis comes just a week after Storm Ciara beat parts of the country, bringing winds of up to 97 mph.
Met Office meteorologist Matthew Box said interior gusts could reach 50 mph by Saturday afternoon, reaching 80 mph in the exposed areas of the south and west.
He said: "The winds will be a landing on what we saw with Ciara but the focus is on a little more rain with Storm Dennis because of the persistence of the rain. "
Higher locations could have 50-80mm of precipitation over the weekend, as high as 140mm in some areas, said Box.
More broadly, rain totals could be around 20 to 40 mm, with the Environment Agency urging people to check for flood warnings.
He said preparations were underway to set up flood defenses, flood storage tanks and to set up temporary barriers to protect communities.
This includes the Foss Barrier in York, the Thames Barrier in London and another in Bewdley, Worcestershire, on the River Severn.
Caroline Douglass, responsible for the floods at the Environment Agency, said: "Remember never to drive or cross flood waters, only 30 cm of running water is enough to move your car - this is not worth not the risk. "
Box said the start of the semester for some children this weekend means families should plan ahead or consider delaying their trip.
He warned that rain could help knock down soggy trees, affecting power lines, while strong winds could affect transportation links such as ferries.
AA urged the public to follow safety advice on the road, with Ben Sheridan, AA Patrol of the Year, saying, "Be very careful when passing high-end vehicles, cyclists and motorcycles , and watch out for sudden gusts, which can blow debris, trees and even damaged vehicles on your way. "