The move follows a North East Linconlshire Council decision to demolish six nearby high rise blocks, moving hundreds of people out.
One of the buildings, Nelson House, was only allocated to older people who now need new accommodation.
Under the plans, Shoreline Housing Partnership would develop 46 one and two bedroom homes.
A group of residents in Nelson House have requested they be rehoused in the same area and together.
The proposal would allow for that and will go before councillors on Wednesday. It has been recommended for approval by council officers.
The development would start on site in early summer and be completed in spring 2016, ahead of the demolition of the East Marsh blocks and Comber Place maisonettes.
A small section of the 115-bay car park would remain. North East Lincolnshire Council said it was currently "greatly underutilised".
A council-owned car park in Grimsby is set to be sold and turned into an older people's housing scheme.