Woodfield Primary School and Grove Road Community Primary School will become one educational establishment in September 2022 if the proposals are approved.
The planned move comes after Woodfield Primary was judged inadequate by Ofsted in January 2020.
Inspectors said pupils at the school were being “let down” by poor leadership and that “too many pupils do not achieve what they are capable of.”
At a monitoring inspection in June, inspectors said improvements were being made at the 56-pupil school.
Any council-run school which is rated inadequate by Ofsted is required to become a sponsored academy.
But no sponsor could be found for Woodfield, and North Yorkshire Country Council (NYCC) instead have put forward the plans for a merger with Grove Road, which they say is an “exciting opportunity.”
If the proposals go ahead, from next September nursery aged children will attend the site at Woodfield, which will be re-named Grove Road Nursery.
Eventually all Key Stage One pupils will be taught at the current Woodfield site, with Key Stage Two being taught at the larger Grove Road site.
A public consultation into the plans will begin next month.
Patrick Mulligan, NYCC’s executive member for education and skills said: “We believe this proposal, which was developed by members of the schools’ governing bodies and our officers, provides an exciting opportunity to enhance the facilities of Grove Road School, organising sustainable education across both sites.
“We are confident it will achieve the best possible provision for all pupils, and importantly ensuring that the Woodfield site continues to be used for education of the wider community.”