Crime author John Harvey is among those who have signed up, taking to Twitter to encourage others to follow suit.
Nottinghamshire County Council is the first in the East Midlands to set up an organisation to manage its libraries, archives, learning and cultural services across the county.
Inspire is a Community Benefit Society which will give people a greater say in the running of their library and other cultural and learning services. A total of 4,046 people signed up in the first week - with that number continuing to rise. The organisation will officially launch in spring 2016.
John is most famous for his series of Nottinghamshire-based Charlie Resnick novels which have formed the basis for a number of events he has taken part in libraries throughout the county.
He hosted a Crime Café event at Sutton-in-Ashfield Library which was attended by book lover Ray Lynn who has also signed up to Inspire.
Ray said: “I enjoy events such as the monthly Crime Café. The county council is definitely going in the right direction with Inspire.
“Hopefully lots of local people will support Inspire – libraries and other cultural services are vital for ommunities.”
Councillor John Knight, committee chairman for culture, at Nottinghamshire County Council said: “We are delighted with the number of people who have become an Inspire member in the first week and it is great to get the endorsement of author John Harvey. Becoming an Inspire member is an exciting opportunity to give people even more of a say in the future management of their local library and other cultural services such as arts and employment programmes.
“We have a proud record of running modern, popular and accessible library and cultural services across the county. We lend more than three million books from our libraries each year, with e-books and e-magazines increasingly popular and host thousands of events.”