Some residents were forced to move out of their homes there after flooding in April.
The Scottish Environmental Protection Agency has warned residents in the capital that two areas could suffer severe flooding.
And flood groups have been formed in Edinburgh and East Lothian amid fears that rivers could burst their banks.
There is also flooding elsewhere in Scotland, while the weather conditions have caused accidents on the roads.
It is expected that if the rainfall continues at the current rate the Water of Leith will burst its banks on Monday evening, causing flooding in the Murrayfield area.
Some residents were forced to move out of their homes there after flooding in April.
People living around Kirkliston can also expect flooding as the level of the River Almond continues to rise.
The heavy rain in the Lothian and Borders Police area prompted the creation of two flood groups.
The Edinburgh group was called into action at 1730 GMT on Monday after SEPA warned the force about rising flood levels.
Co-ordinated by the police, it also features representatives of City of Edinburgh Council, the fire service, East of Scotland Water, Forth Ports Authority and Lothian Health Board.
The River Esk in Musselburgh and the Braidburn in Edinburgh were not at high risk levels, although this could change if the rainfall continues into the night.
Flood group chairman Chief Inspector Grainger Falconer said: "Our information is that the water levels are high at several rivers in the Lothians and Borders but it is the Water of Leith at Murrayfield and the River Almond at Kirkliston that is causing most concern.
"We are asking householders to make preparations to prevent or limit damage to their houses."
He said the heavy rain was expected top continue, although the warnings remained a precaution at this stage.
A similar flood group has been set-up in Haddington to cover the East Lothian area.
Minor injuries
Elsewhere, police in Dundee, Fife, Angus and Perth have all reported localised flooding.
And two people escaped with minor injuries after a tree fell on two cars in near Tillicoutry.
The heavy rain led to flood alerts across Britain as weathermen predicted some areas would get more than two inches in the next 24 hours.
The situation is most serious in North Yorkshire and the south-east, where many towns and villages have flooded again.
More flooding is expected across the country on Tuesday as another band of heavy rain starts to spread from the west, striking the whole of England and Wales.