Negotiations over the planned move are progressing well, according to the council's leader Councillor Alan Mosley.
The authority has been in discussions with Shropshire Council in an attempt to reach an agreement to relocate its main offices to the historic Castle Gates library since last November.
Under the proposal the town council is also looking to take on the management of the town's library service.
Councillor Mosley said: "We are in discussions with Shropshire Council about co-location in the library.
"Shropshire Council is in agreement with that and believe with some adaptations the co-location would be a big success so the council agreed we will continue to work on those lines and hope that could happen quite quickly."
The council needs to find new accommodation as its current base, the Guildhall at Frankwell Quay, is the main centre for Shrewsbury's new university centre.
Councillor Mosley said that an agreement would protect the future of the library service in the town, with Shropshire Council looking to stop managing all of the county's libraries.
He said: "We are in discussions with Shropshire Council about the town council taking over the library services, managing the library services and once again there is agreement and common ground and we are moving forward on those discussions.
"Part of Shropshire Council's intentions is to withdraw from delivering library services and the active operation of it, and it seems possible they will delegate the library function to the town council.
"Our intention is to try, wherever possible, to protect and promote services for Shrewsbury residents."
Councillor Mosley said that some work would be required to allow the council to base its offices at the building, but that they also hope to be able to host council meetings at the venue.
He said: "We hope we can move in by the summer and obviously there is sufficient accommodation space there both for our headquarters and also for potentially for community and town council meetings.
"There will be adaptations necessary to fit our needs and also to ensure our area is fully accessible.
"It fits our requirements for a town centre location. It is a hugely important building which is Grade I listed and we would play a role in its conservation and ongoing viable use."
The library was the home of Shrewsbury School between 1550 and 1882 with a library opening at the site in 1885.
It was temporarily relocated to Raven Meadows in 1976 while the building underwent extensive renovations.
It was officially reopened by Princess Margaret in 1983.