Kaylie Hatton, 16, left the Sidcup address in Bexley on 16 May. She is believed to be in the company of 51-year-old Fred Finch from Eltham.
Scotland Yard said the teenager went missing in the early hours after arguing with her grandmother.
The Met said they had classed them as "high-risk missing persons".
Kaylie and Mr Finch were last seen travelling on a 96 bus in the Welling/Crayford area.
Det Sgt Graham Scott said: "We have had some unconfirmed sightings over the past week and we think they might be using swimming pools and leisure centres to grab a shower.
"The last proof of life we have got is Sunday night between 4pm and 8pm at a friend's address."
Det Sgt Scott said Kaylie had long blonde hair that may have been dyed dark.
She was last seen wearing a white top with a gold chain around the collar, black leggings and tan Timberland boots.
Mr Finch is described as white with short dark hair. He was last seen wearing a baseball cap, a black jacket, black jeans and black Nike trainers.
A "vulnerable" teenager remains missing more than a week after climbing out of a window at her grandmother's home in south-east London.