The EU has written to Theresa May to assure her MPs about the Brexit deal ahead of a crunch vote in the House of Commons – but warned that nothing about it can be changed.

Jean-Claude Juncker and Donald Tusk, the presidents of the Commission and Council, said the EU wanted to do a trade deal by 2020 to make sure the agreement's controversial backstop clause would never have to come into force.

But they said they were “not in a position to agree anything that changes or is inconsistent with the withdrawal agreement”.

In his section of the letter Mr Tusk said earlier reassurances on the deal drawn up by the European Council in December had "legal value", but stopped short of formally committing the EU to signing a deal by a certain date.

Mr Juncker also repeated in writing a pledge he made in December to start preparing for negotiations on a future relationship “immediately after the signature of the withdrawal agreement” in order to maximise the chance of doing a deal quickly.

Behind the scenes in Brussels officials do not believe the latest letter will do much to convince sceptical MPs to vote for May's deal. Conservative eurosceptics, who are set to rebel en masse on Tuesday, want real changes to the withdrawal agreement to give the UK the ability to leave the backstop whenever it wants – or a time-limit on it.

Though both the UK and EU say the backstop ought never be used, in reality it is likely to become the de facto relationship between the UK and EU for many years, because the prime minister's proposals for frictionless trade – a requirement to replace it – have been rejected. She has produced no alternative plans and rejected those suggested by the EU.

Labour says it cannot back the plan because it does not meet its tests of keeping the benefits of single market membership and a customs union. It has also criticised the backstop in its current form.

Trade deals take years to negotiate and most trade experts say it is unrealistic to expect a deal to be concluded before the backstop comes in at the start of 2021. In the letter, the EU chiefs assured Ms May that the transition period could potentially be extended to prevent the backstop coming into force – though there are limits to how long it can be dragged out for.

The transition itself is not popular with eurosceptic MPs, some of whom have claimed it amounts to "vassal" status. During the period, the UK would have to follow all EU rules, including new ones, with no say in drawing them up.

In her letter to the EU the prime minister had said the “deal is at risk” because of “concerns in the UK parliament about how we are delivering our commitments in relation to Northern Ireland’s border with Ireland”.

She said that worries could be “mitigated by changes to the backstop itself”, naming suggested “proposals for a way in which the UK would choose to exit the backstop unilaterally were it ever to come into force, or for a hard time limit on the duration of the usage of the backstop”.

Speaking in Stoke-on-Trent on Monday morning after the letters were exchanged, Ms May said the EU’’s assurances “make absolutely clear the backstop is not a threat or a trap”.

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

“I fully understand that the new legal and political assurances which are contained in the letters from Donald Tusk and Jean-Claude Juncker do not go as far as some MPs would like,” she said.

“But I am convinced that MPs now have the clearest assurances that this is the best deal possible and that it is worthy of their support.”