Cumbria County Council is considering a proposal for the town council to take over the Main Street building.
It the move goes ahead, the tourist information centre would move in alongside the library service.
There have also been early talks about moving the town’s Allerdale council customer service centre into the site.
The move would increase the opening hours of the library, which would have a self-service machine and be staffed by TIC workers.
It comes after the library’s four employees were accepted for voluntary redundancy during the latest round of county council job cuts.
They are due to leave at the end of March, with the takeover proposed for April.
The town council hopes the proposed transfer would secure the future of all the affected services.
Sheila Brown, town clerk, said: “Working in partnership is definitely the way forward in today’s tough economic climate.
“Cockermouth Town Council feels it’s a win win for the councils and the community.”
The county council would continue to supply stock for the library under the proposal. The library has been operating on reduced hours in recent weeks due to sickness.
If the takeover is accepted, Mrs Brown said, opening hours are likely to increase in line with current TIC hours.
The county council’s Allerdale local committee will consider the proposal on Wednesday.
Members will be asked their views to be considered later by the council’s cabinet.
The two-storey library building was building with funds donated by philanthropist Andrew Carnegie and has been run by the county council since 1974.
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