It is alleged that Allan, 20, reversed over a man in his Audi Q7 at 8.50pm on Friday night. The pedestrian remains in a critical condition with head injuries.
The incident took place in Waverley Road, Portsmouth. Allan is currently on loan at the League One club.
Police have put out an appeal for witnesses but Allan was not arrested.
"The pedestrian is a man from the Southsea area, and was seriously injured and treated by paramedics at the scene and taken to QA Hospital in Cosham," a police spokesman said.
"He was later transferred to the neurological unit at Southampton General and remains in a critical condition."
Allan played for Portsmouth on Saturday and was sent off in a 2-0 home defeat to Notts County in the FA Cup.
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