Councillor Adam McVey has said places where “additional risks” have been highlighted by police could be shut off to protect people and reduce pressure on important services during the crisis.
The warning comes after the local authority was forced to close the path to a popular beauty spot in the Scottish capital after four people had to be rescued from Cramond Island over Easter weekend.
On Tuesday morning, Mr McVey told the BBC’s Good Morning Scotland radio show: “We could and we have closed some spaces, like Cramond Island for example, where there are additional risks that the police have identified and we have responded to that.”
But he insisted there are currently no plans to implement more widespread closures of public spaces in the capital, despite many places seeing a surge in the number of people out and about during good weather over the holiday weekend.
“By and large people have been adhering to the guidance and maintaining social distance when they’re exercising,” he said.
“We’re very clearly saying to people: use common sense, read the guidance, make sure you’re clued up on what’s expected of you when you’re in these places.
“There is enough space in these places for people to get the exercise that is so important for well-being.”
He added: “We’re not envisaging closing parks across the city.
“It’s important that people’s health and well-being is balanced within the context of the pandemic that we’re currently experiencing, so there are absolutely no plans to close our public parks.
“It’s about maintaining that open space and using common sense to maintain a social distance when you’re in that space to keep yourself safe, your family safe and those around you safe.”
Lifeboats crews were called out to rescue three men and a woman who had become stranded on Cramond Island when they were cut off by the tide on Saturday.
After being brought to safety they were fined by police for flouting lockdown advice.