It is believed the nurse's suicide note has left her grieving family furious, with her husband Ben Barboza believed to be favouring an inquiry.
A source close to Saldanha's family was quoted by the paper as saying, "One of the letters, which is the longest, deals with the hospital and is critical in its tone. Needless to say, Ben wants a full inquiry into what happened, and he wants to make sure the truth comes out. Within the letter Jacintha calls into question some of the treatment she received at the hospital."
Scotland Yard detectives are examining the notes, as well as interviewing Saldanha's friends, family and colleagues at the hospital. The family has been provided with transcripts of all three notes.
Saldanha's third suicide letter is more of a practical one which deals with her funeral arrangements.
Senior Labour MP Keith Vaz, who is working on behalf of her relatives, said on Thursday night, "The truth of this matter has to come out for the sake of her family".
Australia's media watchdog has launched a rare fast-track probe into the broadcast of the prank call by a Sydney-based radio station.
As the hoax call sparked global condemnation, the Australian Communications and Media Authority said it was using its special powers to launch a rare "own motion" probe contrary to other cases where it has to wait for complaints.
The radio station's owner Southern Cross Austereo had said earlier that all profits from advertising for the rest of the year would go into a fund for Saldanha's family. It had said it would make a minimum contribution of 500,000 dollars.
The two radio presenters behind the prank call had said they were "gutted and heartbroken" over the death of the nurse.