The woman opened the front door to her home in Russell Hill Road, Purley, on 11 May at about 07:50 BST when she realised a man was outside.
He used his foot to force the door open and entered the property where he first assaulted her, bruising her face, before raping her.
The Met said it was "doing everything we can to arrest the suspect".
Det Insp Keith Ward said "specially trained detectives" were providing support for the woman.
The man is described as white, about 5ft 8ins (1.73m) tall, with short brown hair. He also possibly had a Scottish accent.
"We ask that anyone who recognises the suspect's description contact us as a matter of urgency", police said.
A woman in her 70s has been attacked and raped by a man who forced his way into her south London home.