The incident was one of a number of ugly scenes at Hillsborough during the 1-1 draw with Leeds which was televised on Friday night.
Gloucestershire Police said the man was arrested in Cheltenham shortly before 1pm.
A spokesman said: "The investigation is being led by South Yorkshire Police. The man currently remains in police custody."
Kirkland, who has played for England, was shoved to the ground moments after conceding an equaliser in the 76th minute.
A hooligan ran from the Leeds fans onto the pitch and pushed Kirkland in the face, before running back into the crowd.
The incident was caught on camera by Sky Sports who were broadcasting the derby game, yet the suspect was not arrested until yesterday.