A leisure trust is expected to be appointed to manage the borough's libraries next week.
Wandsworth councillors have been recommended to select Greenwich Leisure Ltd (GLL) to take control of all 11 of the libraries and the Wandsworth heritage service, which runs out of Battersea library.
It comes after Wandsworth and Croydon councils joined forces to put their library services out to tender, which will save the council £500,000 a year.
GLL, John Laing Integrated Services Ltd and an in-house Wandsworth Council bid were short-listed, with GLL deemed to offer the best value.
Croydon Council will hold a separate meeting to decide which of the three bids will win their contract.
GLL will run the service from April 2013 for a period of eight years, if councillors give the green-light.
A total of 146 employees are expected to retain their jobs, with the creation of a full time and part time post too.
Councillor Jonathan Cook, cabinet member for environment, culture and community safety, said it will mean the libraries can remain open.
He said: "Our libraries are among the best in London and we are proud to be looking at ways to safeguard and enhance the service at a time when other councils are considering closures.
"We realise, just like all councils, we have to reduce our spending. Bringing competition to the market place will ensure that our taxpayers continue to receive the best value and best service possible."
York Gardens Library and Community Centre, in Battersea, is expected to still be ran as a direct services organisation with volunteers.
But Sonia Francis Mills, chairman of the Friends of York Gardens, said she is concerned GLL will not abide to this promise.
She said: "We have had that reassurance from the local authority but new brooms might well want to do something differently.There is a concern from the charitable arm that the agreement may not be respected.
"My personal view is that libraries are a local authorities responsibility and I think that it is not a great move to remove it away from the public to the private sector.
"The agenda of the private sector is bound to be really different."
GLL is a charitable social enterprise, formed in 1993 to run seven leisure centres in Greenwich.
Since then it has expanded and manages various community services and spaces across the UK.
Councillors will discuss the contract at the environment, culture and community safety overview and scrutiny committee on Wednesday November 15.