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 Ltd to keep them open.
The 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.
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: "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 the 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.
Council accountants say the pools are not energy-efficient and would cost £1.45 million 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.
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.