Thomas Proctor was given leave from HMP Castle Huntly near Dundee but failed to return and was reported missing on Wednesday.
It is thought the 33-year-old, jailed for a knife attack in 2014, may have travelled to Airdrie, North Lanarkshire.
Police have appealed for any information on his location but warned the public not to approach him.
Proctor is around 6ft with short dark hair.
A force spokesman said: “Police Scotland is appealing for information regarding the whereabouts of 33-year-old prisoner Thomas Proctor, who has failed to return to HMP Castle Huntly, by Dundee, following a period of home leave.
“Proctor was reported missing at 3pm on Wednesday June 22.
“If seen, members of the public should not approach him and should contact Police Scotland on 101.
“Anyone wishing to provide information anonymously can do so through the charity Crimestoppers on 0800 555 111.”