The mayor of the Sydney council area taking in Bondi beach said she was in discussion with the New South Wales health minister’s office on Monday about allowing access to the water, as some beaches further south opened for exercise.

As of Monday morning, Randwick council has allowed people to exercise on the sand and in the water on their beaches, which include Coogee, Maroubra and Clovelly.

These were originally closed after large crowds flocked to nearby Bondi beach, ignoring physical distancing directives, in the early weeks of the outbreak.

“Since the beach closures occurred, the government has introduced new social distancing and group gathering [rules],” Randwick mayor Danny Said said in a statement.

“I need to stress that our beaches are not open to general use and can only be accessed for exercise. Activities such as sitting on the sand, sun-baking or gathering in groups will not be permitted and council staff and police will be enforcing these restrictions.”

The Randwick council area is a hotspot for Covid-19 community transmission.

It has the third most community transmissions in the state, 18, and at least 86 cases overall. As a result, testing criteria in the area have been expanded by NSW Health, with the chief health officer urging anyone with Covid-19 symptoms to get tested.

But Said said beach users would be able to maintain a safe distance.

“I’m confident that with the support of swimming and surfing communities that we as a community can all help each other to practise social distancing, stay safe and still enjoy our beaches for exercise,” he said.

Previously it was suggested that Randwick, Waverley and Woollahra councils would be in lockstep, closing and eventually opening all beaches at once to avoid large numbers of people descending on one area.

Randwick is the only council to reopen its beaches.

The Waverley mayor, Paula Masselos, told Guardian Australia she had contacted the NSW health minister, Brad Hazzard, and would discuss plans to allow access to the water this week.

Masselos said on Monday there were no plans to reopen Waverley beaches, including Bondi, Bronte and Tamarama.

“I wish to reiterate that Waverley council has no intention of reopening its beaches at this time,” she said in a statement.

“We are, however, in constructive discussions with the minister for health, Brad Hazzard’s office on providing managed safe access to the water for the the purposes of ocean swimming and surfing for exercise, and are working towards a plan to provide this access.

“We continue to have a significant number of people visiting our area, and whilst this is the case, we need to be more cautious about the approach we take on our beaches.”

To prevent people from travelling long distances to visit its beaches, Randwick council will not reopen car parks. There are also plans to keep the fences in place, limiting access points to the beach.

The council said it might limit numbers on the beach during busier times and people could view all three beaches on live webcam feeds to see how many people were there before they left home.

The ocean pools at the beaches are covered by statewide federal health orders and therefore will not be reopened.

Some have expressed their concern on social media.

“First day back on Maroubra beach at 6am there was a boot camp there with 6 people in it. Great, ruin it for everyone else. They were sharing kettlebells,” one person commented on a Randwick council Facebook post.

“Ok, let’s not stuff this up. Do the right thing otherwise they will put the restrictions back on,” another person commented.

Coogee local Carolyn Martin, who spearheaded the community movement to reopen Randwick’s beaches for exercise, said she was delighted to swim again after three weeks, but hoped people will not travel from afar to exercise.

“I guess that is a concern for the council and the local community, we are hoping that doesn’t happen. We’re hoping that Waverley council can open their beaches, and then that will allow everyone to stay within their communities,” she said.

“Let’s hope there are not too many hot days and people swarm to the beach because it might be hard to control.”

Waverley remains the most heavily infected area in NSW.

Despite Waverley beaches being closed a handful of people have still attempted to swim, some using rocky outcrops to avoid walking on the heavily patrolled sand.

Tim Murray, the head of Tamarama surf lifesaving club, said off-duty lifesavers had had to rescue swimmers caught in rough seas near the rocks.

“For people who aren’t strong swimmers that can be quite dangerous,” he said. “Some people feel that those rules don’t apply to them and so they find areas where there’s less monitoring to go for a swim.”

But Murray said this wasn’t necessarily a reason to reopen beaches.

“One of the problems with reopening in a limited way is: will people comply? Or will they just do whatever they want?”

A spokesperson for Woollahra council said it had no plans to reopen its beaches – situated on the harbour rather than the ocean – in the near future.

“Our swimming locations are much smaller and access at many points is far narrower than at neighbouring beaches, making social distancing very difficult to achieve,” they said.

“We will review our position in the coming weeks.”