The call, which is said to have involved a patient from Wakefield, was described as "unacceptable" by one local councillor.
It comes amid concerns telephone appointments are becoming "the norm" within the health service, at the expense of face-to-face contact with medics.
Wakefield Clinical Commissioning Group (CCG), which oversees NHS services in the area, said all of its practices were offering face-to-face appointments as well as telephone calls.
They said more than half of the appointments provided in July had been delivered in person by doctors, rather than remotely.
The issue was raised at a meeting of Wakefield's health scrutiny committee last Thursday.
Labour councillor Charlie Keith, who represents the Wrenthorpe and Outwood West ward, told the meeting: "A growing number of people have approached me with regard to telephone appointments being used all of the time.
"I fully understand that they were used during Covid and I understand why, but people are concerned that it’s becoming the norm.
"It’s happening from consultant level to GP appointments level.
"This week I had somebody who was told they had 12 months to live by telephone.
Coun Keith added: "It’s completely wrong the way it was done. It’s unacceptable.
"There seems to be no training or protocol being done, so that that kind of news is not handed down by phone."
Former nurse and committee chair, Councillor Betty Rhodes, said she shared the concerns, adding: "We all understand why telephone appointments were brought in during the pandemic.
"I think everybody appreciated why they had to be done.
"But at this particular time we have the NHS declaring that it’s open and the government declaring that the NHS is open.
"Our constituents need to know exactly what is open and what they can get.
"It’s a reasonable question to ask."
In its response, Wakefield CCG said demand for its services was far exceeding pre-pandemic levels.
It said it is currently ranked 11 out of 104 CCGs across the country in terms of the number of healthcare appointments it is delivering.
In a statement, the organisation said: "The Covid-19 pandemic has dramatically changed how we live and work.
"The NHS in Wakefield has worked relentlessly to adapt in order to provide care for patients and communities.
"Alongside remote appointments, all practices in Wakefield continue to provide face-to-face appointments.
"In July, 57 per cent of appointments were face-to-face.
"Health services, including general practices, remain under pressure and staff are working extremely hard so please remember to treat them with courtesy and respect."