News of the move comes the day after a survey showed soaring house prices had pushed the average cost of a home up to pounds 100,000 nation-wide and pounds 156,000 in London.
Lambeth is giving away the houses and flats after allowing them to decay over the last 30 years.
The council said it would take 15 years to carry out the pounds 1bn-worth of repairs, including installing central heating in 8,000 properties and new windows in another 14,000.
A spokeswoman said it is consulting tenants and that one option was to hand the properties to housing associations. She said it was not an unusual move and many councils faced the same problem.
"If you had a map of local authorities you will find the same story - that council homes run into disrepair. It is not a situation that is peculiar to Lambeth.
"There was a meeting this week with tenants about the transfer of their leasehold. It is still in the early stages and it would have to go in to consultation," she said.
Tom Franklin, secretary of housing at Lambeth, said nothing would be decided without agreement from tenants. "There are many tenants in Lambeth whose homes are in desperate need of repair," he said. "Lambeth is doing the right thing by explaining the scale of the problem and searching for a solution."
Shelter, the charity for the homeless, said the give-away showed the need for more investment in affordable housing and urged Lambeth to ensure tenants' rights were protected.
Figures from the Land Registry published earlier this week showed house prices rose by an average of 12.28 per cent over the past year, pushing the average cost of a home to pounds 97,616.