The European Commission has delivered a blow to a plan to protect EU citizens’ rights in the event of a no-deal Brexit – warning that it will “not negotiate mini-deals”.

The House of Commons voted on Wednesday for the “ring-fencing” of protections for EU nationals in the UK and British nationals in the EU27 – effectively saying that the part of the withdrawal agreement that concerns them should come into effect no matter what.

The so-called “Costa amendment”, named after the MP who tabled it, instructed the UK Government to try and secure this outcome in the negotiations – dependent on cooperation from the EU.

But a spokesperson for the European Commission told reporters in Brussels on Thursday after the vote:

“The best way to protect the rights of these 4.5 million people concerned is through the withdrawal agreement. We will not negotiate mini-deals because negotiating such mini-deal outside the withdrawal agreement would imply that the negotiations have failed.”

The spokesperson added: “It’s not for the Commission to comment on amendments voted in the House of Commons where the debate is still on-going, but what I can tell you more generally on citizens’ rights is that the Commission has consistently made clear that the rights of EU citizens in the United Kingdom and UK nationals in the EU are our top priority.

“They should not pay the price for Brexit, and the Commission has called on member states to take a generous process to UK nationals that are already in their territory.”

The European Commission also already outlined a number of ways it would deal with British citizens’ rights in the event of a no-deal – covering social security cooperation, the Erasmus programme, and making sure British nationals do not need short-term visits to visit the continent.

Police separate clashing Brexit protesters outside Parliament Show all 11 1 /11 Police separate clashing Brexit protesters outside Parliament Police separate clashing Brexit protesters outside Parliament Anti-Brexit protester Steve Bray (left) and a pro-Brexit protester argue as they demonstrate outside the Houses of Parliament Getty Images Police separate clashing Brexit protesters outside Parliament A pro-Brexit protester argues Getty Images Police separate clashing Brexit protesters outside Parliament Police surround the pro-Brexit protester after he confronted Steve Bray, a pro-European protester Getty Images Police separate clashing Brexit protesters outside Parliament A leave supporter is spoken to by a police officer as he argues with a remain supporter, Steve Bray, outside Parliament PA Police separate clashing Brexit protesters outside Parliament MPs in Parliament are to vote on Theresa May's Brexit deal next week after her December vote was called off in the face of a major defeat Getty Images Police separate clashing Brexit protesters outside Parliament A leave supporter is spoken to by a police officer PA Police separate clashing Brexit protesters outside Parliament Police look on as anti-Brexit protesters demonstrate outside the Houses of Parliament. Getty Images Police separate clashing Brexit protesters outside Parliament Police hold back a leave supporter PA Police separate clashing Brexit protesters outside Parliament A police officer speaks with anti-Brexit protester Steve Bray Getty Images Police separate clashing Brexit protesters outside Parliament Police surround a pro-Brexit protester after he confronted a pro-European protester Getty Images Police separate clashing Brexit protesters outside Parliament Conservative MP David Davies, wearing a gopro camera, speaks to anti-Brexit protester Steve Bray Getty Images

Citizens’ rights groups have said that these proposals are not enough, however, and that there would still be disruption to their lives under a no-deal.