The watch-and-act message issued on Monday is the first to be aimed at specific residents as water levels continue to rise, ahead of an expected peak in about 10 days.
The State Emergency Service said there was a risk to properties bounded by Crescent Street, Plushsbend Road, and the river within the Crescent levee.
It said crews were on the scene and locals were urged to stay away from the area for their own safety.
Premier Peter Malinauskas said a private levee had been breached, impacting about six properties.
"This is something that we continue to monitor," he said.
The first peak in the river level is tipped to reach Renmark around December 14, with daily flows expected to hit about 175 gigalitres.
The peak is expected to reach Mannum, east of Adelaide, around December 27 where levee strengthening and construction efforts are also in full swing.
After a brief settling of water levels, a second peak when flows could exceed 185 GL a day, is forecast to reach the SA border towards the end of the year.
Up to 4000 properties in SA are expected to be inundated, 450 of which are considered permanent residences.
From Tuesday health authorities will begin to relocate 21 of the most vulnerable residents from the Renmark Paringa District Hospital as a precautionary measure.
"These 21 residents are not independently mobile and require the highest level of care and it is important that time is given to ensure relocation is done in a careful, controlled, and safe manner," local health chief Wayne Champion said.
Prime Minister Anthony Albanese and SA Premier Peter Malinauskas visited Renmark on Saturday to help reassure locals and monitor preparations.
Mr Albanese praised the character and resilience of those in the flood's path who were working together to prepare.
"This is a difficult time but I have been really heartened by the optimism of this local community and just the sheer confidence that this community can be resilient and can get through this," he said.
On Wednesday a second emergency relief centre will open at Mannum where locals impacted by the flooding can access a range of services including personal hardship grants, rental assistance and emergency accommodation, food relief and financial counselling.
A similar centre is also open at Berri in the Riverland.
Meanwhile, preparations for the flooding continued with work beginning on Monday installing temporary flood protection systems in various parts of Berri, Barmera, Renmark, Murray Bridge and Mannum.
"We have confidence that the works undertaken will protect the community and key infrastructure from the impacts expected from the River Murray floods," SES Chief of Staff Graeme Wynwood said.
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