Theresa May’s plans for a new generation of grammars will not mean a return to “sink schools” for children who miss out on selection at the age of 11, a senior Cabinet minister has insisted.

The Prime Minister told a private meeting of Conservative MPs yesterday evening that she wanted to create a “21st century education system” with an “element of selection”.

Labour accused her of advocating “social segregation” in schools, while the government’s social mobility tsar, Alan Milburn, warned that a return to grammars could be “a social mobility disaster”.

But defence secretary Sir Michael Fallon - whose own constituency of Sevenoaks in Kent was the location of effectively the first new grammar school opened in 50 years in 2015 - insisted that the government’s aim was to provide a choice of high-quality schools in every part of the country.

“We need to widen choice,” Sir Michael told the BBC Radio 4 Today programme. “In my constituency I have two excellent academies, I’ve got one of the new free schools and my constituents also now have access to grammar schools, and that’s the kind of choice I want to see in every part of the country.”

Proposals to be announced “very shortly” would offer parents and children across England “a proper choice of good schools” and not “a choice between passing the 11-plus and failing it and having to go off to a sink school of the kind that’s let our children down so badly”, he said.

“Not every child is suited to a purely academic grammar school education and it’s really important that there are proper alternatives that are equally outstanding in the quality of education they deliver.”

Ms May told the 1922 Committee of backbench Conservatives that “selection by house price” already existed within the state school system, with wealthier parents able to ensure a place for their children at high-performing schools by buying homes in the catchment area.

Ms May’s comments could be interpreted as an attempt to calm fears that lifting the ban on new grammar schools will only benefit children of the rich, at the expense of the disadvantaged.

She told the meeting: “We have already got selection, haven’t we - it’s called ‘selection by house price’.”

But Mr Milburn told The Guardian: “Frankly, I still remain sceptical about the social mobility dividend.”

The former Labour cabinet minister said: “This is not selection educationally, it is selection socially. If (more of) that is what is being talked about, it will not provide a social mobility dividend, it will be a social mobility disaster.”

Mr Milburn recommended a number of policies to address inequalities in education, including better pay and discounted housing for teachers who move to disadvantaged areas, having a Ucas-style system for pupils going down the vocational route, and improving parenting skills.

Labour has accused Ms May of favouring an education system that will only cater for the “select few” while unions suggested her plans are elitist.

Sir Michael Wilshaw, the government’s outgoing chief inspector of schools, said on Monday that a return to the selective model would be “profoundly retrograde” and warned the claim that poor children benefit from grammars was “tosh and nonsense”.

Shadow education secretary Angela Rayner said: “Tory education policy can be summed up in one word - shambolic. The Prime Minister talks about social inclusion while at the same time advocating social segregation through grammar school selection.

“No child’s life chances should be defined by a test they sit at the age of 11. The priority for our education system should be investment to raise standards, not investment to create social exclusion.”

Ms May’s comments came after an official was photographed carrying a document signed by the Department for Education’s most senior civil servant, revealing proposals for a consultation on opening new grammars.

The document said education secretary Justine Greening’s “clear position” was that they should only be approved once ministers have worked with existing selective schools to show that pupils who do not make the grade are not disadvantaged.

It also revealed apparent fears that ministers might find it difficult to get legislation allowing new grammar schools through the House of Lords. The prospect of a new wave of grammar schools under a Theresa May government was first revealed by TES in July.

In an effort to allay concerns that new selective schools may damage social mobility, last night the Ms May told the 1922 Committee she wanted new grammars to be “inclusive and not exclusive”.

But Liberal Democrat education spokesman John Pugh said: “There is no such thing as inclusive grammar schools. By their very nature they exclude children who don’t pass a test aged 11. We want every child to succeed, not just the few.”

Former deputy prime minister Nick Clegg said Mrs May had no “mandate” for a return to selection.

He told Today: “I know they talked about grammar schools in their manifesto. They didn’t talk about extending selection to free schools and so on, which seems to be now what they intend to do.

“This is the danger of having a new government with new leadership and new priorities, not elected by the people, now foisting their own evidence-free prejudices upon us. There’s no evidence at all that that is the answer to many of the problems in our education system.”