The county council said 110 of the region’s 359 schools will welcome back reception, year one and year six classes on that date.
A further 24 are planning to reopen from June 8.
But coronavirus safety concerns over small classrooms and staff shortages means some schools will stay shut.
The county council said it could not yet confirm which schools are reopening.
It comes after weeks of wrangling between the government and teachers’ unions – and some 50 councils elsewhere in England are believed to be ready to defy the government’s reopening instructions.
North Yorkshire County Council has backed the government’s plan and is assuring parents that children’s safety will be a priority.
In some schools children will be taught in small ‘bubbles’ – spending the day with the same, small group of other children and teacher.
Staff changes will be minimised and items touched by children will be sanitised.
Regular hand-washing will also be in place.
Coun Patrick Mulligan, the council’s executive member for education, said: “Not all schools will be open on June 1; we are a large rural county with a relatively high number of small rural schools – including traditional church schools based around small classrooms – where implementing social distancing simply isn’t possible.
“We would like to reassure parents who are sending their children to school on Monday that schools are working hard to ensure the safety and well-being of pupils.”
Union officials at NASUWT, the UK’s second-largest teaching union, had previously said the reopening plan is “seriously at odds with scientific evidence” and “the deep concerns expressed by schools, teachers and parents.”
The council added it is working with all other primary schools to reopen in the coming weeks.