A campaign by Labour activists to abolish private schools has been endorsed by John McDonnell as a move to make it official party policy gathers pace.

In a statement this morning, the shadow chancellor and close Jeremy Corbyn ally said he backed the campaign “to talk about how we ensure an integrated education system, where private schools don’t need to exist, and should not exist”.

I hope this campaign will pick up support and eventually become Labour Party policy

The campaign, called AbolishEton, has prepared a motion to be put to the Labour Party conference this weekend. It has the backing of the Momentum group of party members, so organisers are confident it will be selected for a vote.

Should the vote pass in Brighton, it will become Labour policy to withdraw private schools’ charitable status, business rate exemption and “all other public subsidies and tax privileges”, and redistribute their endowments, investments and properties “democratically and fairly across the country’s educational institutions” if the party wins a general election.

“I hope this campaign will pick up support and eventually become Labour Party policy,” said McDonnell.

“I think we can gain a large number of votes on this issue, because I believe people think that everyone should have a fair start in life, and that starts by making sure that we all have the same access to education facilities.”

This weekend’s conference motion can still be amended and watered-down during a process called “compositing”, during which delegates putting forward the motion meet with the shadow education team, led by Angela Rayner.

But Holly Rigby, co-ordinator of the campaign, said she was “really hopeful that it’s not going to be watered-down”.

“We know Angela has been really committed to securing a socially just and equal education system, and we want the most radical version of our demands that we can possibly get,” she said.

The motion is now backed by more than 350 Labour councillors and 17 Labour MPs, but not everyone is as supportive.

Private school leadership groups are unsurprisingly against the idea, and have warned of an increased burden on taxpayers, and that the best state schools will become more elitist due to rising house prices.

Mike Buchanan, executive director of the Headmasters’ and Headmistresses’ Conference, said: “It is frustrating to see yet another ill-thought-through political attack on families who are already saving the state thousands of pounds each a year and ask for no rebate on their taxes.

“In some cases, part of the fees paid by these parents will already be used to fund their schools’ charitable activities and additional support to state schools.”

Dr David James, a private school deputy headteacher and former adviser to Conservative schools minister Nick Gibb, warned on twitter that the proposal would lead to the closure of “many” schools, to the detriment of staff and pupils.

He demanded that the National Education Union and NASUWT “unambiguously speak out in support of all [independent school] members against the political threats to our schools & pupils”.

Responding, Dr Mary Bousted, the joint general secretary of the NEU, said her union “will enter into discussions with Labour as and when policy announcements are made about independent schools in order to ensure teacher and support staff jobs are protected in all circumstances”.

Rigby said insisted unions were “on the side of private and state school teachers”, and said she and other NEU members would propose a motion at the union’s conference next year, aimed at making it official NEU policy to support the abolition of private schools.

“We’re very clear we’re not talking about knocking private schools down or closing them down, we’re talking about integrating them, so private school teachers just become state school teachers and actually have much fairer conditions. We see now that private school teachers are about to go on strike over their pensions. They’re not well-protected like we are to some extent in the state sector.”