Donald Tusk is to ask Theresa May for “a better idea that would be as effective in preventing a hard border” on the “island of Ireland”.
The prime minister rejected the EU’s proposals on Northern Ireland saying they would split the UK in two.
But the European Council president said one of the “possible negative consequences” of the kind of Brexit Mrs May wants would be a hard border.
The two are meeting at No 10 ahead of a big Brexit speech on Friday by Mrs May.
Mrs May, who chaired a meeting of the cabinet before her talks with Mr Tusk, has already pledged not to accept the draft withdrawal treaty published on Wednesday by the EU.
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The UK and the EU both agree on wanting to avoid a return to a physical border – with border posts and checks – in Northern Ireland.
The UK has suggested new IT systems could be introduced to avoid the need for physical border checks but has yet to spell out how this would work in practice.
But the draft EU treaty also includes the option of a “common regulatory area” after Brexit on the island of Ireland – in effect keeping Northern Ireland in a customs union – if no other solution is found.
Both the EU and the Irish government say it is up to the UK to come up with concrete alternatives to what they describe as a “backstop” option.
Mrs May said the EU proposal would “threaten the constitutional integrity of the UK” by creating a border down the Irish Sea.
But in a speech in Brussels, Mr Tusk said he was “absolutely sure that all the essential elements of the draft” would be accepted by the 27 remaining EU members.
And he said Mrs May’s decision to rule out membership of the single market and customs union had been acknowledged “without enthusiasm and without satisfaction”.
The PM has said she wants a deal which will allow trade to be “as frictionless as possible”.
But Mr Tusk warned: “There can be no frictionless trade outside of the customs union and the single market.
“Friction is an inevitable side-effect of Brexit by nature.”.
Irish senator Neale Richmond, European affairs spokesman for the Fine Gael party that leads the government, said Britain has provided “zero detail” on its proposed alternatives to keeping Northern Ireland in a customs area with the EU to avoid a hard border with the Republic of Ireland.
Veteran Eurosceptic Sir Bill Cash told the BBC’s Newsnight programme there were “technical ways” of managing the Irish border and accused the EU of trying to create a “constitutional crisis” for the UK.
Cabinet ministers have suggested Friday’s speech by Mrs May will give the EU the clarity that it has been seeking about what kind of trade relationship the UK wants after its departure on 29 March 2019.
In an apparent concession to the EU ahead of the speech, the government said EU nationals coming to the UK during a transition period after Brexit, expected to last two years, would get indefinite leave to remain.
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Mrs May has said her long-term goal is a “bespoke economic partnership”, underpinned by a comprehensive free trade agreement guaranteeing tariff-free access to EU markets for British goods and services.
But her predecessor Sir John Major warned on Tuesday that an “a la carte entrance” to the European market was not possible if the UK left the single market and customs union – which Mrs May is committed to doing.
Losing existing trade advantages, he said, would make the UK a less attractive place for inward investment and could put 125,000 jobs at Japanese firms at risk.
Conservative Brexiteers have criticised Sir John’s intervention, in which he held out the possibility of another referendum on the final deal, one describing it as “un-statesmanlike” and full of “cheap comments”.
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But the message will be echoed by former Labour prime minister Tony Blair later in a speech in Brussels.
He will call for the public to have a “real choice” while urging the EU to put forward new ideas to address “genuine underlying grievances beneath the Brexit vote, especially around immigration”.
In a new report, the Commons Business Committee warned failure to reach any kind of deal would be damaging for the car industry and only close alignment with the EU would ensure its survival.
But on Tuesday the industry received a vote of confidence when Toyota said it would build the next generation of its Auris hatchback at its Burnaston plant in Derbyshire, safeguarding more than 3,000 jobs.