Under budget proposals for next year, the council would reduce its grant to the arm’s- length organisation by a total of £3.35 million over the next four years.

Liberal Democrat councillor Kevin Lang said: “We have looked very closely at the figures and we struggle to see how they can make savings on that scale without closing down facilities.

“And given the focus on trying to encourage a more healthy and active lifestyle, that’s the last thing we should be doing.”

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Edinburgh Leisure faces a £3.35 million funding reduction. Picture: Lisa Ferugson

The cuts would see Edinburgh Leisure’s funding reduced by £350,000 next year and £1m in each of the following three years.

The council’s budget document says: “In 2019/20 a modest reduction in grant is proposed to Edinburgh Leisure which we expect them to absorb through internal efficiencies. There is the potential for more significant savings in future years through new partnership to deliver local services.”

Tory councillor Andrew Johnston said the cuts could mean price hikes, reduced hours and centres being amalgamated.

“I’m very concerned that Edinburgh Leisure appears to be in the firing line,” he said.

“Sports centres play an important role in helping people keep fit and many people across the city use them to stay healthy, both physically and mentally. If these centres are under threat it’s only going to put more strain on the health and social care budget, which is already under severe pressure.

“When people have had time to digest these proposals I think people will be vociferously opposed to them.”

Green councillor Gavin Corbett said he understood the proposed grant cut was about passing Edinburgh Leisure the responsibility for funding pay rises and also creating economies of scale by expanding the scope of what it oversees.

“But I know there will be very close attention to any hint that swimming pools, sports centres or pitches might be lost.

“As a rapidly expanding city, we desperately need more opportunities for exercise, not fewer.”

Last year the council scaled back plans to hit Edinburgh Leisure with a £420,000 funding cut and a £375,000 maintenance charge for sports pitches after public objections and fears of closures.

And in the run-up to the 2015/16 budget, Edinburgh Leisure chiefs warned that swimming pools and sports venues could close because of a proposed £2m budget cut.

It was said at the time that, apart from flagship venues such as the Royal Commonwealth Pool and the Edinburgh International Climbing Arena, almost no pool or sports centre was safe.

Among the venues thought to be under threat were the leisure centres at Ainslie Park and Gracemount, several bowling greens and Dalry Swimming Pool.

The council is having to find total savings of £41m from next year’s budget. Proposals were published on Friday, ahead of a finance committee meeting next week.

Council leader Adam McVey said Edinburgh Leisure’s saving in the first year should be “fairly easy” for them to deliver.

And he said they needed to “ramp up” options for savings in the following three years

“Some of them will take time to deliver – some of them will be about better using some of the facilities that they have got.

“I would hope that Edinburgh Leisure would come back to us next year with options that actually look at enhancing their business model. I think we’ve given them the time and space to look at it far more innovatively.

“The overall budget of Edinburgh Leisure in terms of all their revenues is around £40m so a £3.35m reduction forecast over those four years in that context is about eight per cent.

“A business that has facilities all over the city, that has a very diverse client base with more facilities than anyone else in the city, has an ability to drive up a business demand and replace some of that.

“They may come to us next year and the following years with proposed additional savings. My hope is they meet the challenge we have set them in driving up the revenue of the business.