Independent firm Making Waves is pitching to Glasgow City Council to take over St Bride's pool in Govanhill, and Abercorn pool in Cowcaddens.
But the move would mean the council paying Making Waves £20,800 for pupil access to the venues.
As first reported in the Evening Times last year, four pools, including St Bride's, Abercorn, Lorne Street Primary, in Govan, and Ashcraig, in Garthamlock, were at risk.
The pools are among 11 run by Education Services in Glasgow, and range in age from 40 to 110 years old.
The council claims the pools are rundown with, among other problems, "tile breakages, sediment problems, leaks, broken drainage pipes, problems with filtration and incidents of personal injury".
Number-crunchers say the pools are losing money and would need around £1.45million to bring up to scratch.
Lorne Street pool has been closed since April 2014 due to a boiler failure, with an estimated cost of £20,000 for replacement
Making Waves Limited stepped in with an offer to lease the pools and run them as a commercial enterprise.
If the 10-year lease is signed it will give Making Waves control of the pools and responsibility for all repairs.
Councillors will discuss issues around the move at a meeting of the Children and Families Executive Committee this week.
These include making sure pupils still have access to the pools, legal duties placed on the headteachers, security arrangements and the renaming of the venues.
St Bride's pool would be known as Strathbungo Pool, and Abercorn pool would be known as Garscube Pool.
Making Waves would charge the council £20,800 each year for pupils to use the pool.
Parents and locals have been battling to save St Bride's Primary School pool, one of four to be axed by Glasgow City Council.
The St Bride's pool is used by pupils and also by local swimming groups.