The two schools are 10 minutes' walk from each other.
Jess, whose son is in reception at Moat Primary, said the school means a "hell of a lot" to her family.
"My mum's worked here for 20 years, I came here as a pupil," she said.
"All my siblings came here, I always thought I could see him just staying here until he goes off to secondary school. It's just a shame."
A consultation on the proposal is already open, and concludes on 6 December.
If approved, Moat School would close either from 1 September 2025 or 1 September 2026.
"We have many vacancies at Moat, we have a small number at Robinswood," Mr Smith continued.
"I would be surprised if parents didn't respond in a heated way.
"We're talking about children's lives and education, but we always want to give people a chance to ask questions, and we're open as a trust," he added.