Children aged 11 could sit exams usually taken by 15-year-olds under plans put to the Scottish Executive.
Pupils could be sitting Intermediate One at primary school
East Renfrewshire Council wants primary seven pupils to be allowed to sit Intermediate One, the new equivalent of general level Standard Grade.
It said many children are bored in class and do not reach their potential.
The radical proposal is in response to the executive's request for feedback on its draft plan to change tight constraints on when children sit exams.
In March, Education Minister Peter Peacock announced the move to relax the rules preventing children moving ahead of their classmates.
He said: "Every child is unique and that is why a 'one size fits all' approach to education is simply inappropriate."
Mr Peacock said he believes Scotland's school curriculum and exams system must be flexible enough to help all children achieve their full potential.
East Renfrewshire, which has the best results in Scotland, said the regulations had to change.
The council is suggesting that primary seven pupils take the Intermediate One exam in subjects such as computing and practical science.
James McVittie, the head teacher of St Ninian's High School in Giffnock, led the council's working party on the issue.
He said: "We have a system in which young people have been told you are in primary seven, that's first year work and you can't do it, or you are in third year and that's fourth year work, you can't do it.
"What we know is that individual young people are moving through their courses each at his or her own pace and that they are well capable, in individual circumstances, of meeting the challenges of courses at higher levels."
However, Dr Brian Boyd of Strathclyde University said exams were not necessarily a good way to motivate bright children.
Education Minister Peter Peacock believes in flexibility
"The end effect of having examinations which were designed for 16 or 17-year-olds moved into upper primary is that they might not be appropriate and they might narrow the curriculum.
"The tendency with examinations is for teachers to teach towards the test, and that is entirely the wrong thing to import into the primary.
"If they want challenge there are lots of ways of challenging young people using thinking skills, formative assessment, creativity and so on.
"We really don't need more exams in the system."
Any changes to the rules would come into effect next year.