The numbers are broadly in line with the Scottish average and have remained at a similar rate over the last 10 years.
The latest report into children's weight also shows around 1.5% are at risk of being underweight.
In Glasgow those at risk of obesity and being overweight was 21.7%, a small increase on the previous year from 21.3%.
Those at risk of being overweight went up from 10.8% to 11.4%, while the rate of obesity fell from 10.5% to 10.3%.
The changes are not enough to be statistically significant.
Children who live in the most deprived areas were found to be at a greater risk of being obese, but not overweight and the researchers could not establish any link between deprivation and children being underweight.
Across Scotland the rate of obesity and overweight children in primary 1 was 21.3% which is similar to the figure in 2002, when it was 22.3%. Over the period it has fluctuated between 20.9% and 23.3%.
The report stated: "The level of inequalities observed are greater for children at risk of obesity compared to children at risk of overweight.
"In 2012/13 in the least deprived areas, 6.6% of children were classified as at risk of obesity compared to 11.6% in the most deprived areas.
"In the least deprived areas 11.2% of children were classified as at risk of overweight compared to 12.8% in the most deprived areas."
In North Ayrshire, the level was far higher than the Scottish average. Despite a modest drop, the rate of obesity and overweight primary 1 children there was 30.4%
In Glasgow 1.3% were classed as underweight, up slightly from the previous year. Across Scotland 1.2% were underweight with those in deprived areas more at risk but not always.
The researchers stated: "The relationship between deprivation and the propor-tion of children in primary 1 at risk of underweight is less clear.
"Figures over the last decade show the prevalence of underweight tends to be higher in the most deprived areas, although this pattern is not observed every year. In 2012/13, no clear relation-ship between underweight and deprivation was evident."
The report by Dr Rachel Wood, consultant in public health medicine stated: "There is continued concern over the levels of overweight and obesity among children in Scotland.
"Obesity during child-hood is a health concern in itself, but can also lead to physical and mental health problems in later life, such as heart disease, diabetes osteoarthritis, back pain, increased risk of certain cancers, low self esteem and depression.
"Underweight in child-hood can also be a cause for concern, indicating poor nutritional intake and underlying medical problems. Both develop as a result of an imbalance between energy consump-tion and expenditure."