Claims by Theresa May that grammar schools have proved to be great engines of social mobility have been undermined by figures showing that the vast majority of grammars admit only a tiny proportion of children from the poorest families.

As pupils return after the Christmas break, official figures show that in many grammar schools, less than 1% of the total pupil intake receives free school meals (an indicator of the poorest families). Only one of the 163 selective schools in England takes in more than 10% from this least well-off group.

Critics of May’s plans to expand the number of grammar schools – one of her few firm policy announcements since entering No 10 last July – say the figures expose her central justification for the controversial plans as entirely bogus, as better-off families tend to monopolise school places at the expense of the poor.

An official consultation on the government’s plans for more grammars closed before Christmas and a bill to take forward the proposals is expected to be announced in the Queen’s Speech in spring – despite large numbers of Tory MPs and much of the House of Lords being strongly opposed.

Analysis of the official data by former Labour education spokeswoman Lucy Powell shows that, on average, just 2.6% of children in grammars are on free school meals, compared with 14.6% in secondary schools across the country. Eligibility for free school meals is calculated according to family income, both earned and from benefits.

Sir William Borlase’s grammar school in Marlow, Buckinghamshire, had, in 2016, just three pupils on free school meals (0.3% of its pupil total), and many others did little better. More than 20 other grammars had fewer than 10 pupils who met the criteria and at well over half, less than 5% of pupils were from poorer families.

Analysis by Powell’s team shows that if children on free school meals were to be admitted to grammars in the same proportion as those going to non-selective schools – in those parts of the country where there is selection – grammars would have to admit 13,500 more such children.

“The idea that grammar schools are engines of social mobility is compete nonsense,” said Powell. “They are the enemy of social mobility for the many. There are pitifully few children from low-income backgrounds at grammar schools. All the evidence shows us that bright children from all backgrounds do just as well at good comprehensives.

“Ministers should focus on spreading excellence across the whole system, ensuring there are enough excellent teachers and that head teachers have sufficient resources.”

As part of their consultation, ministers have sought views on how grammar schools can increase the number of lower-income children they serve.

Powell plans to table amendments to any bill to ensure that the plans give far greater opportunity to children from less well-off families.

Critics of the prime minister’s plans also dispute claims that children from disadvantaged families perform better in selective schools than in non-selective ones.