The Eton figure represents 3.5% of the £10.54m spent by the government on UK private schools last year.
The boarding school subsidies are for top diplomats who may be posted abroad at short notice.
The Foreign Office says it wants to ensure children don't have their education disrupted if their parents have to leave the country.
In the same year, £21.38m was spent on school fees for 902 UK diplomats' children in education overseas.
Under the longstanding scheme - called the Continuity of Education Allowance - senior Foreign Office staff can apply to have up to 90% of their children's school fees paid by the taxpayer if they contribute the remaining 10% themselves.
A total of 531 children had their education subsidised at UK schools in 2021/22.
In the last academic year, the maximum amount that could be claimed for secondary pupils to board at schools was set at £11,814 per term, equivalent to £35,442 per year.
Ten children subsidised to attend Eton College last year did so at a total cost of £371,827 - an average of £37,182, about 5% higher than the maximum allowance.
We asked the Foreign Office why the average amount spent at Eton appears to be higher than the cap but they did not give an explanation for the discrepancy.
The figures were obtained through Freedom of Information requests by Labour's shadow attorney general Emily Thornberry.
James Roberts, managing director of the TaxPayers' Alliance campaign group, said: "Hard-pressed taxpayers will feel put out by these figures.
"While it's understandable that diplomats want the best education for their kids, we don't need to subsidise sending their children to our most elite public schools.
"Diplomats should expect to send their children to one of our brilliant state schools, like everybody else does, or pay private school fees themselves."
The maximum amounts in 2021/22 diplomats could claim for children included £11,814 per term for "senior boarder" fees, £8,270 for "senior day school" fees, £10,385 for "junior boarder" fees, and £7,629 for "junior day" fees.
Labour's Freedom of Information request showed the amounts spent by the government at the 10 UK private schools where the most places were funded in 2021/22.
These included £629,073 spent on 23 pupils at Sevenoaks School, in Kent, and £381,851 on 11 pupils to Oundle School, in Northamptonshire.
In its current form, the allowance dates back to 1996.
A Foreign Office spokesperson said: "In order to represent UK interests across the world, our diplomatic and development staff can be asked to move frequently during their career, sometimes at very short notice.
"It is our longstanding policy to support eligible families with education, to ensure that children do not suffer any disruption during key points in their academic development, and there are clear caps in place on compensation offered to staff that are reviewed regularly."