The body of Denis Cannon, 55, was found at HMP Glenochil. He was serving a six-year sentence after he was convicted of attempting to kill himself and his partner by feeding a hosepipe into his car in a layby in Brechin.
Sentencing him in 2010, Judge Lord Bannatyne told the High Court it was an "extremely nasty offence".
John Perry, 24, who was on remand at HMP Perth in connection with drugs offences, was taken to Perth Royal Infirmary on the same day, where he died.
Police and next of kin have been informed of the deaths and they will be reported to the procurator-fiscal.
A fatal accident inquiry may be held in due course.