by Maureen Gitau, 24, was reported missing on 10 December, having last been seen five days earlier as she left her home in Evelyn Street in Deptford, Lewisham.
Police believe Ms Gitau has come to serious harm although a body has not been found.
A 54-year-old man, thought to be known to Ms Gitau, was arrested on suspicion of her murder on Friday.
Det Ch Insp Kate Blackburn, said: “This arrest is a significant development in what is now being treated as a murder investigation.
“Whilst we have not located Maureen’s body we have reason to believe she has come to serious harm and has likely been killed.
“Maureen’s family have been informed and they will be supported by specialist officers.”
Her family reported her missing and have made appeals for information.
Police are asking anyone who knew Ms Gitau and who saw her on 5 December in the Deptford area, or anyone who believes they have information that could assist their investigation, to call 101.
