The 82-year-old was killed when she was struck on Leigh Road, Leigh, at about 11:45 GMT on Wednesday.
Police said that after the crash, officers were told a man "was seen to steal her purse before leaving".
The man is described as white, about 6ft (1.8m) tall, between 30 and 40 and of medium build with a brown beard.
Det Sgt Neil Lawless said it was "one of the most disgusting crimes I have ever investigated".
"I find it hard to believe that someone could stoop so low to commit such a horrific offence," he said.
I don't understand why stealing a dead woman's purse is "one of the most disgusting crimes I have ever investigated". It seems to me that if you are going to assign relative degrees to theft and make it context sensitive, that stealing something from someone who obviously no longer needs it has less impact than from someone who does. I just don't understand the moral relativity.