Officers have released a CCTV image of a man they would like to speak to in connection with the incident.
Police were called to Glengarnock Avenue junction with Manchester Road, E14 at around 6am on Sunday, 12 May to reports of a woman assaulted at home.
The victim reported that she was asleep in bed when she was woken up by the male suspect carrying out a sexual assault.
The woman managed to run out of the room and the culprit left soon after via the front door.
Officers investigating the incident believe the same man was seen in another flat in the block just moments before.
Again, the occupants woke up to find the man in their bedroom, who then turned and left before he could be confronted.
The suspect is believed to be a white male, in his 20s, and at the time was dressed in black, wearing a black coat with a white collar and white trainers.