The incident took place on the stretch of pavement between Tesco Express and the traffic lights at the junction of Grove Road North and Elm Grove, at 3.20pm on Sunday.
The elderly man collided with the cyclist - described as being a man of large build, wearing a top hat – who did not stop to check on the man and verbally abused him as he cycled away.
The pensioner sustained several injuries, including fractures to his hand, wrist, and arm.
A spokeswoman from Hampshire Constabulary said: ‘Officers believe there may have been a number of witnesses in the area at the time and we would like to hear from anyone who heard or saw what happened.
‘We would also like to hear from anyone with dash cam or Go Pro footage from the area at the time.’
Anyone with information is asked to call 101 and quote reference number 44210096761.