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European Council president Donald Tusk hinted at a frosty meeting with Theresa May as he told the Prime Minister he is "not happy" with her Brexit "red line".

The top Brussels official met with Mrs May in 10 Downing Street at one of the most fractious moments in Britain's negotiations with the EU.

Their talks came a day after the Prime Minister rejected EU proposals to establish a "common regulatory area" between the bloc and Northern Ireland after Brexit.

As he sat alongside Mrs May in Number 10, in front of photographers, Mr Tusk told the Prime Minister: "I'm absolutely sure that after your so-called red line, we'll I'm not happy with it, you know, but of course, but it's natural that you have maybe different point of views when it comes to the essence of Brexit.

"Anyway, after your decision on no customs union and no single market it's some kind of breakthrough and we can start our substantive negotiations immediately."

Mr Tusk also apologised for his lateness to a lunch of poached lemon sole, followed by chocolate pavlova, joking that his "transport was not frictionless" as the UK is gripped by freezing temperatures.

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Earlier in the day, the European Council President had dismissed Mrs May's hopes for post-Brexit trade between the UK and EU to be "as frictionless as possible".

An EU source described Mr Tusk's meeting with Mrs May as "an open and honest debate in a good atmosphere about the real political difficulties ahead of us".

They said the "main focus was on the content of, and process towards, the future relationship post-Brexit" with the European Council president having taken note "of the repeatedly stated UK red lines and recalled that the red lines will shape the future relationship".

Their talks also touched on the planned Brexit transition period and Northern Ireland, while Mr Tusk confirmed all other 27 EU member states were giving their "full support" to the bloc's chief Brexit negotiator Michel Barnier, the source added.

A Downing Street spokesperson said Mr Tusk and the Prime Minister "held a positive and constructive meeting", where they looked forward to the European Council summit later this month.

Mrs May was "clear" the UK is "fully committed" to turning December's agreement on the first phase of Brexit negotiations - on key divorce matters - into a legal text "as soon as possible", and is "steadfast" in her commitment to avoiding a hard border on the island of Ireland, the spokesperson added.

However, the Prime Minister was also said to have told Mr Tusk the draft legal text presented by the EU on Wednesday was "unacceptable" due to its impact on the "constitutional integrity" of the UK.

Mrs May also briefed the European Council president on the contents of her major Brexit speech on Friday and expressed her hope EU leaders "would engage with this thinking constructively", the spokesperson added.

On Thursday morning, in a speech in Brussels, Mr Tusk said the EU acknowledged Britain's "red lines" to quit the bloc's single market and customs union "without enthusiasm and without satisfaction".

But he added the EU would "treat them seriously with all their consequences".

"There can be no frictionless trade outside of the customs union and the single market," he said.

"Friction is an inevitable side-effect of Brexit. By nature."

He also revealed he would be asking the Prime Minister for "a better idea" than the EU plan she has rejected for avoiding a hard border on the island of Ireland.

"Until now, no-one has come up with anything wiser than that," Mr Tusk said.

Mr Barnier also called on Britain to provide "better ideas" on how to avoid a hard border between Northern Ireland and the Republic of Ireland, "while preserving the integrity of the single market".

Speaking to a business conference in Brussels, the European Commission official added: "We are ready to look at them in a very constructive way."

Mr Barnier also warned Britain the EU "cannot and will not compromise on its founding principles" during negotiations on a future EU-UK relationship.

"I recognise the political temptation to avoid making a choice or to downplay the cost of Brexit, or indeed to pretend that the UK could obtain a free trade deal with the EU with all the benefits of the single market without the obligations," he said.

"Abandoning such ideas will enable us to begin building an ambitious future partnership based on the foundation of realism and in the interests of our citizens and our businesses, and this is my objective."

He also warned "certainty" on the Brexit transition period may only come "early next year".

On Wednesday, Mrs May insisted "no UK prime minister could ever agree" to a plan for a "common regulatory area" between the EU and Northern Ireland, included in a 118-page document set out by Mr Barnier on Wednesday morning.

The proposal would see such an area established in the event a EU-UK agreement on a future relationship fails to prevent a hard border on the island of Ireland.

Under the plan, Northern Ireland would remain an effective part of the EU's customs union, align with the EU on VAT, abide by Brussels rules on state aid, and remain under the jurisdiction of the European Court of Justice.

But the Prime Minister insisted the EU scheme, if implemented, would "threaten the constitutional integrity of the UK" by creating a border in the Irish Sea.

It has also sparked angry accusations that Brussels is trying to "annex" Northern Ireland.

On Friday, Mrs May will flesh out the Government's position on the future EU-UK trade relationship ahead of the second phase of Brexit negotiations.

It follows a two-hour Cabinet meeting on Thursday, where ministers were described by Downing Street as having "agreed on the speech as being a real step forward" after a "genuine discussion ahead of the text being finalised".

Next week, Mr Tusk will set out the EU27's own guidelines on a post-Brexit trade agreement.