This would see the school’s current site in Lea village closed and its 117 pupils forced to travel into Malmesbury for classes.
Wiltshire Council has identified that schools in Malmesbury and the surrounding areas are currently unable to meet demand and more places for pupils must be found.
A letter from Lea and Garsdon’s head teacher Kay Vousden and the chair of governors David English was sent to parents to explain the options.
It said: “The first would be to expand the current school in Lea to accommodate the expected rise in pupil numbers, this would create a one form entry school with seven classes (eventually with 30 pupils in each year).
"The second proposal is looking at a piece of land at Filands in Malmesbury, intended to be used to build a new primary school. This would see Lea and Garsdon CE primary school relocating to this site, again as a full one form entry school.
“The proposals are purely at the feasibility stage, one of gathering information, however the governing body feel it is appropriate to share this information with you at this very early stage, to fully include the school community.
“There are a huge number of questions to be answered and a large amount of information to be gathered, before any proposal can be properly considered and the governing body will ensure that the whole school community is kept informed as this process develops.
"Should either of the two options being explored be considered feasible there will then be a formal process of consultation this year with all those who have an interest.”
John Parmiter, parish clerk of Lea and Cleverton parish council, reacted to the news the school could close with sadness.
“The feeling in the village is that we are very much opposed to the closure as it would tear the heart out of the community,” he said.
“It would be a great loss, we’d only be left with the pub and the church if the school closed.”
A Wiltshire Council spokesman said: “We always have to look to future school provision to ensure there are sufficient places so that children can attend a school near where they live.
“At this stage we have commissioned feasibility studies to explore options to increase primary school places in this area.
“We are not currently consulting on any proposals, however when our work to look at the options available is complete we will consult with parents and the local community.”