It follows the revelation that cash-strapped chiefs at Scottish Borders Council are planning to merge the service with its contact centres in a bid to save money.
However, it is understood that the libraries in Galashiels and Hawick will not be affected by the proposed move.
Gordon Edgar, chairman of The Royal Burgh of Selkirk and District Community Council, said: "I understand that librarians are being made redundant to accommodate this and the library will be run by trained contact centre staff."
But he added: "If they are going to do it in one town they need to do it in every town otherwise we want a reduction in our community tax."
Councillor Vicky Davidson, a member of the local authority's ruling administration, pointed out the council needed to save money. She insisted a library service would be maintained in every town in the Borders - but she confirmed "there may not be as many librarians".
Mr Edgar warned: "I fear that the library service is going to be depleted to such an extent that it won't be retained in smaller towns like Selkirk."