Cumbria County Council’s Allerdale local committee has supported the proposal, which would see the tourist information centre move into the Main Street building.
But concerns have been raised about the effect the changes would have on the library service.
The county council has now agreed to consult on the plans.
A spokesman said a consultation document would be produced, with people invited to send their views or attend a drop-in session in the town.
Town, borough and county councillor Eric Nicholson said there would be an opportunity to share views at an already-planned meeting to discuss library opening hours.
Coun Nicholson said the meeting will be held in Cockermouth, on a date to be confirmed.
He added: “When two or three concerns were raised about the transfer of operations to the town council is was decided, as part of the consultation into opening hours, we would also encourage people to come in and voice their concerns about the other changes which we can then maybe address.
“Some people think this is the end of line but it’s not. We are being forward thinking.
“We can explain how it is going to work. I’m fairly confident that once it’s properly explained how the library will operate in the future and how it has been operated in the last 12 to 18 months, people will realise that it is the way forward.”
The county council’s cabinet will make a final decision early next year.
The town council said the move would increase the opening hours of the library, which would have a self-service machine and be staffed by tourist information centre workers.
It comes after the library’s four employees were accepted for voluntary redundancy during the latest round of county council job cuts.
There have also been early talks about moving the town’s Allerdale council customer service centre into the site.
The town council hopes the proposed transfer would secure the future of all the affected services.
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