I had a look at their official web-site, and came up with the following information...
By phone +44 (0) 8712 710 711*
(Please note that the switchboard is closed between the hours of 2100 and 0600)
*Calls are charged at 10 pence per minute from a BT landline. Calls from mobiles and other networks may vary
The phone number of the Manchester Library is: 360-871-3921.
The phone number of the Manchester Public Library is: 860-643-2471.
The phone number of the Manchester District Library is: 734-428-8045.
The phone number of the Manchester City Library is: 603-624-6550.
The phone number of the Manchester Historical Society is: 978-526-7230.
The phone number of the Manchester Historical Society is: 860-647-9983.
The phone number of the West Manchester Branch is: 937-678-8503.
The phone number of the Manchester Branch is: 732-657-7600.
The phone number of the Diocesan Museum Of Manchester is: 603-624-1729.
The phone number of the North Manchester Public Library is: 260-982-4773.
The phone number of the Manchester-By-The-Sea Pub. Library is: 978-526-7711.
The phone number of the Manchester Historic Association Libr is: 603-622-7531.