The Rotherham Institute of Obesity was set up after a healthy eating campaign in the town by TV chef Jamie Oliver.
It is one of 21 public health services being reviewed by the council after a Â£1.3m budget cut from the government.
Clinical manager Dr Matt Capehorn said: "Any degree of funding cuts means that we financially can't afford to operate anymore."
The population of Rotherham is 259,000 and 73% of adults in the town are overweight and obese, according to authority figures.
Dr Capehorn said: "We're very concerned for the population of Rotherham.
"We see on average about 1,500 patients referred to [the institute] each year and our budget is Â£300,000 from the local authority... Even a small financial cut means we would have to close."
The council said the cuts were "deeply regrettable".
"The fact that we're losing Â£2m over this three-year period will affect that and the council has no choice but to reduce those services," said Councillor Chris Read.
MP for Rotherham Sarah Champion urged the authority to "not cut this service".
"This is literally saving lives in [my] constituency," she said.
The Department of Health would not comment on the specifics of Rotherham's funding.
The clinic was commissioned by the NHS in 2009 after Oliver launched his Ministry of Food project to encourage people to increase their knowledge of food and how to cook.
Oliver targeted Rotherham after mothers were pictured shoving burgers and chips through school railings in protest at menus suggested by him in a Channel 4 programme.
A clinic tackling obesity claims it may be forced to close because of cuts in funding.