The board of management of Mount Cara said it was committed to keeping the Redemption Road facility open until January 2021 "if a solution is not found".
Cara House is a 25-bed home, with 15 current residents, which opened more than 30 years ago.
The board of management said last month that due to Covid-19 and a reduction in occupancy, it was to close.
Staff insisted they had been told in early June that the facility would stay open, only for a reversal last month, and had vowed to fight to keep it open.
Community vigils have taken place weekly outside the facility, with staff members being joined by families of residents and local representatives.
Families and staff also called on the HSE to step in and save the facility, which the health body says it cannot do.
According to company accounts for 2019, with statements approved by the board of directors on April 28, Cara House saw a surplus of €22,467 for the financial year.
It is in receipt of funding from Cork City Council, the HSE, and residents' fees, on which it says it is "fully reliant".
Board of management chair, Fianna Fáil city councillor Dr John Sheehan, said: "The board is acutely aware of the financial pressure Mount Cara is under and will work with the HSE to try and secure extra funding while the board explores each and every option re Mount Cara.
"However, it is important to note that this funding is not secured at present, although the HSE has been supportive so far."
He said the board has "worked tirelessly over the past number of months to assess each option" in relation to Mount Cara’s future.
"This has involved extensive engagement with the HSE, HIQA, Fair Deal, along with the resident’s families, the community, and other interested parties. While discussions with interested parties are at early stage, they have positive so far and the Board will continue with these engagements," he said.
Minister of State at the Department of Health with Special Responsibility for the Elderly, Mary Butler, has offered her support, he said, as did the Taoiseach's office.
"While the outcome for Mount Cara is still uncertain, the pathway to a solution for the residents of Mount Cara is laid out," he said.
Sinn Féin TD Thomas Gould said the board has listened to calls for time "to prevent the closure of this important facility".
“I have now received a positive response from the HSE promising support in keeping the facility within the community. What’s clear today is that the great people of the Northside of Cork won’t be silenced," he said.
He criticised Taoiseach Micheál Martin for not personally intervening, as well as health minister Stephen Donnelly.