Police have twice arrested the man, who has written threatening letters and made phone calls, but each time he has been released. An assessment by psychiatrists has found he is not sufficiently disturbed to warrant being held under the Mental Health Act.
Parents at Liskeard Junior School have been told every precaution is being taken to protect their children.
Their fears have been heightened by the killing of 16 children and their teacher by crazed gunman Thomas Hamilton at Dunblane Primary School, Scotland, in March, and the machete attack at St Luke's Church of England School in Wolverhampton in July.
Chair of the school governors, Clive Ellacott, said: "We do not believe the threat to be large, but we have to take it seriously."
Val Cox, chair of the education committee, said: "We are now in a Catch 22 position: the police cannot act until he does something, and we are using the guards to ensure he doesn't."
She said the bill for the operation would run into thousands of pounds and would present problems for an LEA already working under budget constraints.
The nature of the threat is not being divulged at the request of police, who are aware of the man's whereabouts and that he has a link with the school.
Next week Lord Cullen is expected to publish his long-awaited report into the circumstances leading up to and surrounding the Dunblane tragedy, putting school security and the vetting of people with access to school premises in the spotlight.
Teachers' leaders have been campaigning for increased funding for improving school security and have accused the Government of an inadequate response following a spate of incidents in and around schools.
Following the fatal stabbing of London headteacher Philip Lawrence last December, ministers set up a working group to study school security and have promised a series of measures.