Donald Tusk has warned after the second big defeat of Theresa May’s deal that he expects a credible reason for any delay to Brexit.

Moments after the prime minister announced that the House of Commons would vote on an extension to the article 50 negotiating period beyond 29 March, the European council president issued an EU red line.

“Should there be a UK reasoned request for an extension, the EU27 will consider it and decide by unanimity,” a spokesman for Tusk said. “The EU27 will expect a credible justification for a possible extension and its duration. The smooth functioning of the EU institutions will need to be ensured.”

A spokesman for the European commission president, Jean-Claude Juncker, echoed Tusk’s position in a coordinated statement.

There is frustration in Brussels at the failure by Downing Street to lay down any groundwork over a potential extension, raising the risk that leaders could reject any request.

At an EU-Arab summit in Sharm el-Sheikh last month, the prime minister had broached the issue of an extension of a few weeks during a meeting with Tusk to allow legislation to go through the Commons should her deal be ratified. But she declined to engage in any further discussion of the options if the deal failed again in the Commons, leading officials to remark that there had been “no charm in Sharm”.

After losing by 149 votes, the fourth largest defeat ever on a government motion, May nevertheless told the Commons she would allow a free vote in her party on an article 50 extension. She said such an extension would be short, and that it risked a new cliff-edge in June, suggesting the British government is looking at a three-month delay.

The EU’s 27 heads of state and government are set to discuss any request next Thursday afternoon at a leaders’ summit on 21 March in Brussels.

Juncker had said that the Commons was being given a “second chance” when he unveiled a package of assurances over the temporary nature of the backstop on Monday evening in Strasbourg, adding that there would be “no third chance”.

A letter published at the same time from Juncker to Tusk stated that in the event of an extension request the UK’s withdrawal should be complete before the European elections on 23 May as British MEPs would otherwise need to be elected.

But officials representing Tusk informed ambassadors on Monday that they believed an extension until July was still feasible as the European parliament would not have convened until that point.

Sources suggested that the European commission, as “guardian of the EU’s treaties”, was simply preparing the ground for a formal infringement notice on the UK should the government not organise elections but that 23 May was not the outer limit of an extension.

The EU has become increasingly concerned that the British government is seeking to pin blame for a no-deal Brexit on Brussels.

Across Europe, leaders and senior officials expressed their dismay at Tuesday’s result. The EU’s chief negotiator, Michel Barnier, said the “impasse can only be solved in the UK”, adding that the EU’s “no-deal preparations are now more important than ever before.”

The Dutch prime minister, Mark Rutte, tweeted that he expected a “creditable and reasoned” justification should an extension request be made, reiterating Tusk’s line: “The smooth functioning of the EU institutions needs to be ensured.”

Spain’s prime minister, Pedro Sánchez, tweeted of his “regret” at the Commons decision.

The European parliament’s Brexit coordinator, Guy Verhofstadt, said the UK had “spiralled out of control”. Edgars Rinkēvičs, the Latvian first minister, said he hoped “common sense will prevail and we will find a way out of this rather difficult situation”.

A spokesman for Tusk said the EU was “disappointed that the UK government has been unable to ensure a majority for the withdrawal agreement agreed by both parties in November”.

“On the EU side we have done all that is possible to reach an agreement”, the spokesman said. “Given the additional assurances provided by the EU in December, January and yesterday, it is difficult to see what more we can do. If there is a solution to the current impasse it can only be found in London.”

The spokesman went on: “The EU for its part continues to stand by the withdrawal agreement, including the backstop, which serves to prevent a hard border in Ireland and preserve the integrity of the single market unless and until alternative arrangements can be found.

“With only 17 days left to 29 March, today’s vote has significantly increased the likelihood of a ‘no-deal’ Brexit. We will continue our no-deal preparations and ensure that we will be ready if such a scenario arises.”

Elmar Brok, a German MEP close to the chancellor, Angela Merkel, said the EU would not accept a long delay to Brexit because London “doesn’t have a clue”.