Linsey Cotton, 33, was jailed for three years in October after admitting fraud and threatening Margaret and Nicola McDonough.
The women died in hospital after being found seriously injured in a hotel in Greenock in May 2013.
Police said they were investigating the death of a prisoner at HMP Saughton.
Paisley Sheriff Court heard that Cotton tricked Michael McDonough, 33, out of Â£5,000 and then tried to extort Â£5,500 from his mother Margaret, 53, and her daughter Nicola, 23, from Paisley.
She told the women they faced 20 years in jail for breaching a non-existent confidentiality clause.
A Police Scotland spokeswoman said: "Police in Edinburgh are investigating after the death of a 33-year-old woman prisoner from HMP Saughton on Sunday morning.
"A report will be sent to the procurator fiscal."
Fatal accident inquiries are routinely held for deaths in custody.
A woman jailed after trying to extort money from a mother and daughter who died in an apparent suicide pact has died after collapsing in prison.