A software "issue" meant polling cards and some postal votes were sent out to some "non eligible" voters.
They are still awaiting responses from six councils so the number could rise.
Leave campaigners Iain Duncan Smith and Bernard Jenkin have expressed "serious concerns" about the "conduct of the EU referendum and its franchise".
They have written to the prime minister complaining of "many and varied" examples of EU nationals being sent polling cards.
On Friday, the Electoral Commission confirmed the numbers affected so far, and said the software provider had since "resolved the issue".
Any postal votes wrongly issued would be cancelled and none of those affected would appear as an eligible voter on the electoral registers used at polling stations, it said.
EU citizens from non-UK countries are not entitled to vote, unless they are citizens of the Irish Republic, Cyprus or Malta.
But Mr Duncan Smith has said a Nottingham City Council officer had emailed a Vote Leave supporter to say the council was unable to check whether people put their correct nationality on an application, and had to assume the information submitted was correct.
Kingston-upon-Thames council in south-west London said a Polish citizen in the area - whose case had been highlighted by the Guido Fawkes website - was sent a polling card because he had put "British" as his nationality on the electoral register.
British, Irish and Commonwealth citizens resident in the UK and many British citizens living abroad are entitled to vote in the referendum.
You can check with your local authority's electoral services team if you are worried that you are not on the register.
If you are not on the register you have until midnight on Tuesday, 7 June to submit an application for a vote. The deadlines for applying for a postal vote at the referendum are:
In England, Scotland or Wales, you can register to vote online anytime at gov.uk/register-to-vote. If you are a British citizen living abroad, you can register to vote online in the same way.
In Northern Ireland, visit the government's Register to vote in Northern Ireland page to download a registration form. If you are an Irish citizen living abroad who was born in Northern Ireland, visit the Electoral Office for Northern Ireland website to download the correct form.
Polling cards were wrongly sent to at least 3,462 EU citizens who are not allowed to vote in the EU referendum, the Electoral Commission has announced.