A woman was walking along Sisters Avenue in the early hours of last Friday when she noticed a man was tracking her.
Another incident happened in Sumburgh Road just off Clapham Common at around 1.30am on Sunday.
This time, the man approached the woman and grabbed her but she managed to escape.
Officers are treating the incidents as linked.
The suspect is described as white, between 25 to 30 years old, of slim build and with short blonde hair.
Chief Inspector Dave Lee, from South West BCU, said: “These were extremely frightening incidents for the women involved and we are using all of the resources at our disposal to identify the individual responsible.
“We know this will cause community concern and we will be carrying out extra patrols at key times. We would encourage anyone who believes they can help our investigation to contact us as soon as possible.”
Anyone with information should call 101 quoting CAD 3425/11OCT. Alternatively call Crimestoppers anonymously on 0800 555 111.