The final choice will be made by Bank of England Governor Mark Carney in 2019.
Mr Carney has already indicated a scientist will be featured on the note.
In a letter to The Sunday Times the campaign calls on the Bank of England to "reflect modern multicultural Britain" by putting a person of colour on a note.
It said: "Ethnic minority communities represent 14% of the British population.
"We do not lack candidates, and arguably their achievements were the greater for having been made at a time when many careers were effectively closed to them, whether through colonial rules, racism, or the legacy of slavery.
"However, no one from an ethnic minority has yet featured on a banknote."