There are fears for the future of Whiteinch Library, which is still closed.
Depute council leader David McDonald said the city would make a “compelling” case to “get as much money as possible” from a £1.25 million pot.
First Minister Nicola Sturgeon announced the money to “help get and keep libraries open” when she revealed her party’s programme for government earlier this week.
Cllr McDonald, who is also the chairman of Glasgow Life, which runs the city’s community venues, was responding to a question from Green group councillor Jon Molyneux.
He had asked how much money would be secured for Glasgow, and what that would mean for the libraries which remain closed.
Cllr Molyneux also said the £1.25m fund “seems to be on the small side to say the least, if it’s to be shared across the country”.
“We know it will cost £1.2m to reopen the final five libraries in Glasgow, and £400,000 for my local library, the Couper Institute,” they added.
Currently, five Glasgow libraries are closed and without a reopening date: Whiteinch, Maryhill, Barmulloch, GoMA and the Couper Institute.
Cllr McDonald said it was “probably too early” to give accurates answers, but talks will take place to “understand just how much of that funding we can secure for Glasgow”.
“It is my intention that we make as strong as possible a pitch to get as much of that money as we possibly can. I would certainly welcome the cross party support of all colleagues in doing that,” he added.
“We will be looking to make the strongest possible case and I would certainly hope that we would be able to include venues and library campaigns across the city in that approach, in making a clear, strong and compelling case of why Glasgow’s libraries are so important.”
Cllr Molyneux also asked when Cllr McDonald had last met with the Scottish Government to discuss further funding for community venues.
The Glasgow Life chairman said there was “regular and ongoing contact with the government at every level”.
He last met with ministers on June 14, and Glasgow Life officers met with Scottish Government civil servants on September 3 about “these related matters”.
Labour councillor Stephen Curran asked Cllr McDonald to engage with campaign groups “who need to know there is a plan even when the final details are not yet firmed up”.
Cllr McDonald said Glasgow Life has regular discussions with campaigners and he would be “more than happy” to hear from them directly.