The floods, caused by the rising waters of the Nile and Gash rivers due to the rainy season, have destroyed hundreds of mud-brick houses in Khartoum.
Authorities fear the Nile will continue to rise, further submerging the lower areas surrounding the capital.
They are also concerned that diseases could break out.
The UN's World Food Programme has launched an emergency relief effort to deliver food to the estimated 20,000 displaced around Kassala, with the UN Mission in Sudan providing helicopters to help deliver the aid.
Egypt has also sent a Hercules plane carrying aid including water purification equipment and tents, Egypt's official Mena news agency reported.