It can cause rashes following skin contact and illness if swallowed.
The algae can also be fatal to wildlife, livestock and pets if ingested.
The council said the fountains in the ponds had been temporarily switched off to reduce the risk of contamination.
Officials would continue to manage water oxygen levels to protect the fish and maintain the health of the ponds, a spokesperson added.
The eastern pond and the Rose Bowl will be monitored and the fountains will be kept on there to reduce the chance of blue-green algae occurring.
Councillor Charles Quinn said: “Ensuring everyone’s safety and wellbeing is our top priority, which is why we have acted quickly following concerns about possible blue-green algae in the ponds.”