Councillor David McDonald wants to make emergency food supplies available in his constituency area of Pollok.
There is already at least one food bank in Govanhill, one in Renfrewshire and one in Inverclyde.
The Evening Times reported last year that people in the north of the city were stealing food from shops because they were in desperate need of food and were then being referred to food banks by police.
Councillor McDonald, who covers Greater Pollok, has suggested some of the city's libraries could house small scale food banks.
He said many of his constituents had been supportive of the idea after the story appeared in the Evening Times on January 3.
However, Mr McDonald said Glasgow Life, which runs the city's libraries, had yet to respond directly about such a scheme.
The councillor is also in early discussions with volunteers and organisations about setting up a new food bank in Pollok.
Social deprivation, high levels of unemployment and poor housing are serious problems in areas such as Nitshill, Priesthill, Arden and Corkerhill.
Mr McDonald said: "I received a number of calls and e-mails from people supporting the idea of food banks in libraries.
"While I have not had any response from Glasgow Life, I am working with local people and organisations in my ward to establish one in the area."
Figures show The Trussell Trust, which aids people across the UK, helps about 7000 Scots, including 1900 children, who have been relying on food hand-outs since April last year.
The charity is expanding its number of food banks in Scotland from 11 to 21.
Benefit cuts, unemployment and rising living costs have led to an increase in the number of people in Scotland using food banks.
Figures show shoplifting incidents soared 14% in the city during 2011-12.
Chief Inspector Ann Hughes, of Maryhill police office, said some of the thieves – including a pensioner caught trying to take three tins of salmon – needed help from charities rather than trouble from the courts.