The decision will affect nearly 200 people living in its facilities at Sylvania in the south, Carlingford in the northwest and its Narrabeen home on the northern beaches.
The imminent closures come after the Christian community organisation shut its Wesley Tebbutt facility at Dundas in northwest Sydney last year.
CEO and superintendent Reverend Stu Cameron said staffing challenges resulting from the national aged care reforms were behind "the difficult decision".
The aged care royal commission made 148 wide-ranging recommendations and the federal government has committed to having registered nurses in aged care homes around the clock from July.
The sector is experiencing chronic staff shortages and the minimum pay rate is due to increase by 15 per cent from July.
"Wesley Mission supports these once-in-a-generation reforms, improving quality for all care users," Rev Cameron said on Thursday.
"But we took the extraordinarily difficult decision - it was time for us to exit the sector.
"There's all sorts of pressures, workforce pressures, cost pressures."
A spokesman for Aged Care Minister Anika Wells told AAP Wesley Mission did not consult the government before announcing the closures, saying the organisation made a business decision.
"The vast majority of residential facilities will meet 24/7 nursing requirements," the spokesman said.
"The Quality and Safety Commission will actively work with facilities which show genuine effort to implement around-the-clock nursing, but (will also) have challenging circumstances.
"Failure to meet 24/7 requirements is not enough on its own for the commission to close a facility."
The sector's peak body called the decision "regrettable but understandable in the current reform climate".
The Aged and Community Care Providers Association says the sector is working hard to deliver quality care, but 70 per cent of providers are operating at a loss, losing an average of $28 per resident per day.
"That is simply not sustainable," a spokeswoman said in a statement.
"The aged care sector is facing enormous financial challenges and workforce shortages, at the same time that it is racing to implement the once-in-a-generation reforms."
Rev Cameron said Wesley's aged care offering was small compared with the diverse range of community services it provides around NSW.
"It is ... a challenging environment to be a smaller provider," Rev Cameron said.
MyCarePath has been appointed to help the 199 residents find a new home before centres close at the end of May.
"We're committed to remaining open until every one of those residents finds a suitable, appropriate new home and care service," Rev Cameron said.
Under government regulations, residents are not required to move until sufficient accommodation has been found.
Rev Cameron said the 249 workers affected would be helped to find a new role within the organisation or with a different provider.
"Wesley Mission's support for older people in NSW continues, focusing on providing in-home care and retirement living to help people stay in their homes for longer," he said.
