Nine unions, representing school leaders, teachers and support staff, have accused the Government of showing a "lack of understanding" about the dangers of the spread of coronavirus in schools.
The joint statement, published by the Trades Union Congress (TUC), warns that staff will "not be protected" by social distancing if primary schools reopen to more year groups from next month.
It says: "We call on the Government to step back from June 1 and work with us to create the conditions for a safe return to schools based on the principles and tests we have set out."
The plea comes after the Government announced its ambition for all primary school pupils in England to go back to school for a month before the summer.
The Government said it expects children to be able to return to nurseries, and for Reception, Year 1 and Year 6 pupils to be back in school, from June 1 at the earliest.
The joint statement, from organisations including the NAHT school leaders' union and the National Education Union (NEU), says: "We all want schools to reopen, but that should only happen when it is safe to do so.
"The Government is showing a lack of understanding about the dangers of the spread of coronavirus within schools, and outwards from schools to parents, sibling and relatives, and to the wider community."
It comes after the Department for Education (DfE) issued guidelines on Monday which said primary school class sizes should be limited to 15 pupils and outdoor space should be utilised.
The advice, on how to safely reopen schools, calls for lunch and break times to be staggered, as well as drop-off and pick-up times, to reduce the number of pupils moving around.
On the guidance, the joint statement from the unions adds: "Uniquely, it appears, school staff will not be protected by social distancing rules.
"15 children in a class, combined with their very young age, means that classrooms of four and five-year-olds could become sources of Covid-19 transmission and spread."
"We do not think that the Government should be posing this level of risk to our society," it says.
It comes after unions sent a list of key measures to the Government last week which they say must be met before pupils in England can safely return to their desks.
The message, issued to the Education Secretary Gavin Williamson on Friday, called for extra money for deep cleaning and personal protective equipment (PPE), as well as local powers to close schools if clusters of Covid-19 infections break out in a particular area.
Paul Whiteman, general secretary of NAHT, told MPs on Tuesday that they had not been consulted on specific plans, announced in recent days, to admit some year groups from June 1.
Kevin Courtney, joint general secretary of the NEU, also advised members not "to engage" with planning for a return next month until further union advice was issued.
But addressing MPs in the House of Commons on Wednesday, Mr Williamson said: "When you have medical and scientific advice that is saying it's the right time to start bringing schools back in a phased and controlled manner, it seems only the right thing to do and the only responsible thing to do."
He added that the Government has worked "very closely with all the teaching unions and headteachers' unions", with time made available each week to discuss matters.
But Mr Williamson acknowledged that allowing pupils to return to school would be "challenging".
He said: "We continue to follow the best medical and scientific advice and believe that this phased return is the most sensible course of action to take."
The statement was signed by teaching union NASUWT, NEU, NAHT, GMB, Unison, Prospect, Unite, the Association of Educational Psychologists (AEP) and the National Society for Education in Art & Design (NSEAD).