A private firm has now stepped in to offer to run the pools, which Glasgow City Council said are too expensive to repair and maintain.
Negotiations could now take place with company Making Waves Limited to keep them open.
Next Thursday's meeting of the Children and Families Executive Committee, councillors will be asked to take steps towards saving the pool.
Those pools under threat include ones at St Bride's Primary School, Govanhill; Lorne Street Primary, Govan; Abercorn in Cowcaddens and Ashcraig, Garthamlock, which range from 40 to 110 years old.
Parents and community members have been battling to rescue St Bride's Primary School pool, one of four to be axed by Glasgow City Council.
The move has come after private firm Making Waves Limited registered its interest in running the pool.
Local councillor Soryia Siddique has been leading the campaign to save the pool.
She said: "I am delighted my campaign to save St Bride's pool has progressed.
"I have been communicating with education, St Bride's parent council and constituents in raising their aspirations for saving the pool.
"It's fantastic that the wishes of the local community, parents, school pupils and pool users are being heard."
The St Bride's pool is used by pupils and also by local swimming groups.
There are no other pools in the immediate area following the closure of Govanhill Baths in 2002.
There are still several steps to be taken before the pool is formally rescued.
The council group will consider whether or not they should refer the report to Executive Committee.
If it is sent forward then committee members will decide if the council should hold talks with Making Waves Limited.
Once talks have been held then a second report will be compiled for councillors to consider, including conditions of the leave and a management agreement.
In January, the Evening Times told how the council planned to save cash by closing four school swimming pools it claims are rundown.
Among several faults, a risk assessment report showed the pools have "tile breakages, sediment problems, loss of pool water, broken drainage pipes, problems with filtration systems and incidents of personal injury".
Council number-crunchers say the pools are not energy-efficient and would cost £1.45million to upgrade.
In May it was decided to give the pools a stay of execution - firstly until October and now until December 31 - with the hope community members would step in to run them.
Parents at St Bride's - backed by Councillor Siddique - kept on with their drive to rescue the pool.
Glasgow City Council currently runs 11 school swimming pools, including seven in secondary schools.
The pools are maintained by Amey for the secondary estate and Cordia (Services) LLP for the others.
Due to investment and maintenance costs and the increased energy and water consumption associated with swimming pools of this age, a proposal to close the four swimming pools was approved by the council on February 7 as part of the annual budget.
At the meeting on February 7, a commitment was given for the education services to look at ways of keeping the swimming pools open. As a result, they will remain open until at least December 31.