Tony Cameron said spot checks at Cornton Vale over recent years had led him to make the assumption.
His views came in a written response to a parliamentary question by an SNP MSP.
It said that while there were no exact figures on arrivals, all new inmates were offered help for alcohol, drug or mental health problems.
Prison service spokesman Tom Fox added: "That is simply our experience of the prisoner group we are dealing with.
"Only occasionally are we finding prisoners who do not have a drug problem."
'Poor records'
Nationalist Stewart Stevenson said he was not surprised by the response to his question, but he called on the service to improve the way it monitored the help being given.
He said: "Although the answer is telling us that everyone is offered help they can't tell us how many took it, which seems rather bizarre when virtually their entire client base is there because of drugs.
"I think the prison service needs to concentrate on that and manage it in much more detail so they can tell us about their successes and failures."
Mr Fox said the service should not be criticised simply because the statistics were not recorded in the way requested.
"That we do not hold these statistics does not undermine the fact that we put a great amount of time and resources into meeting the needs of prisoners when they arrive in prison," he said.
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