A new wave of grammar schools will tackle “selection by house price” and give every child the chance to go to a good school, Theresa May will say this morning.

Selective schools will have to meet targets on how many pupils they have from poor families or set up new open access schools and help failing institutions, under the reforms being set out by the prime minister.

The controversial proposals will also allow other schools to introduce selection “in some circumstances”.

Universities that want to charge higher fees would be made to set up a new school or sponsor an existing underperforming school.

In her first domestic speech since entering No 10, Mrs May will claim that back-door selection by postcode and wealth is “unfair”.

The premier will dismiss Labour’s opposition to selection as sacrificing children’s potential for “dogma and ideology”.

Mrs May will say: “We are going to build a country that works for everyone, not just the privileged few. A fundamental part of that is having schools that give every child the best start in life, regardless of their background.

“For too long, we have tolerated a system that contains an arbitrary rule preventing selective schools from being established - sacrificing children’s potential because of dogma and ideology.

“The truth is that we already have selection in our school system - and it’s selection by house price, selection by wealth. That is simply unfair.

“That is why I am announcing an ambitious package of education reforms to ensure that every child has the chance to go to a good school.

“As well as allowing new selective schools, we will bring forward a new requirement that means universities who want to charge higher fees will be required to establish a new school or sponsor an existing underperforming school.

“This is about being unapologetic for our belief in social mobility and making this country a true meritocracy - a country that works for everyone.”

Mrs May told the 1922 Committee of backbench Conservatives on Wednesday that the current state school system allows wealthier parents to ensure a place for their children at high-performing schools by buying homes in the catchment area.

But the Government’s social mobility tsar, Labour former Cabinet minister Alan Milburn, has said that grammars could be “a social mobility disaster”.

And Labour shadow Angela Rayner warned MPs that an expansion of grammar schools would “entrench inequality and disadvantage”.

Rules stopping faith schools from selecting more than half of pupils according to religion are also set to be relaxed under the proposals.

A Downing Street source said: “The admissions cap had the best of intentions but it has failed in its two key tests. It has failed to make minority faith schools more diverse, because parents of other religions and none do not send their children to those schools.

“But it has prevented new Catholic schools from opening, which are more successful, more popular and more ethnically diverse than other types of state school.

“We’re going to change the rule, so we can allow new Catholic schools to open, while making faith schools of all kinds do more to make sure their pupils integrate with children of other backgrounds.”

Sam Freedman, executive director of education charity Teach First, said: “Even with quotas, by their nature grammar schools will only ever support a small proportion of children and not those who would benefit most from a good education.

“We already know the best ways to use education to support social mobility and we need to focus on making sure all pupils get an outstanding education, leaving school ready for life in a global economy.”

Sarah Pearson, interim director of New Schools Network, a charity that helps free school applicants welcomed the university reforms and the scrapping of the limits on selection by faith.

“We welcome the idea outlined today that would see universities further incentivised to get directly involved in setting up free schools. From Birmingham to Exeter, King’s to Cambridge they have already founded some fantastic schools and it would be very exciting to see a new generation of university schools,” Ms Pearson said. 

“It is also good news that the government will be consulting on re-examining the rules restricting faith schools to 50% of their pupils being faith based. This is something the NSN has called for over the past year and, together with new measures that will ensure genuine integration, will support greater community cohesion.”