NHS Grampian said a total of 42 people who attended Stonehaven Vaccination Centre on 1 February were given a vaccine that was later discovered to be out of date.
The health board said it believed "the doses administered are safe and will still protect".
But the people affected are being offered another jab if they want one.
Those affected are aged 12 years and older.
An NHS spokeswoman said: "We believe the doses administered are safe and will still protect.
"However, there is a very small possibility of reduced effectiveness, therefore we are offering re-vaccination if those affected want this.
"We are contacting them directly to apologise and to offer them this, according to their personal circumstances."
The health board added that its "investigations to date suggest this was an isolated incident and no other centres are affected".