The stretch of the Paraguay River that flows through this capital surpassed critical-flood stage on Wednesday as the number of evacuees approached 7,000.
Asuncion's emergency-management office said that the water level increased 4cm (1.6 in.) in a 24-hour period to reach 5.52m (18 ft.).
More than 1,500 families living in riverfront neighborhoods have been evacuated since the river started rising on Saturday.
While some evacuees went to shelters, many have taken up temporary residence in crude sheds of wood, metal sheeting and plastic that they must often build themselves.
Asuncion authorities have recommended that residents still at risk make plans to evacuate on short notice.
In December 2015, heavy rains associated with the weather phenomenon known as El Niño led to devastating floods in Asuncion that forced nearly 100,000 residents to flee from their homes.