He warned that UK "ambiguity" must be removed and progress on "separation" issues made before any talks on the future EU-UK relationship.

For his part, UK Brexit Secretary David Davis said both sides had to show "flexibility and imagination".

The two men spoke at a brief media call before the talks got started.

The two sides have played down the prospect of a breakthrough in this week's talks.

BBC Europe correspondent Kevin Connolly said the mood had been set by a series of briefings "that betray a good deal of mutual exasperation".

The EU has accused the UK side of "magical thinking" on the issue of the Irish border.

And the UK has said the Europeans are "massively over-egging" their financial demands on the UK in terms of an "exit" bill.

The third of the main "separation" issues to be resolved covers the rights of EU citizens in the UK and UK citizens in the EU after Brexit.

Mr Barnier said: "We need you to take positions on all separation issues. This is necessary to make sufficient progress. We must start negotiating seriously.

"We need UK papers that are clear in order to have constructive negotiations. And the sooner we remove the ambiguity, the sooner we will be in a position to discuss the future relationship and a transitional period.

"The EU 27 (the EU members excluding the UK) and the European Parliament stand united - they will not accept that separation issues are not addressed properly.

"I am ready to intensify negotiations over the coming weeks in order to advance."

Mr Davis said that the UK's goal remained to secure a "mutually beneficial" agreement that works for people and businesses across Europe.

He said this week's "technical talks" would build on the July round and the papers the UK has recently published on its vision for its withdrawal from the EU.

"For the UK, the week ahead is about driving forward the technical discussions across all the issues.

"We want to lock in the points where we agree, unpick the areas where we disagree, and make further progress on a range of issues.

"And we're ready to roll up our sleeves and get down to work again once more."

Mr Barnier said on Twitter last week that the focus of the third round of negotiations would be "orderly withdrawal".

Officials from both sides will meet in working groups to discuss the detail behind each side's proposals, before the round is closed by Mr Davis and Mr Barnier on Thursday.

Setting the scene - political correspondent Chris Mason

Fairly technical. No big bang. No breakthrough. But no breakdown - 10 words that don't make for a great headline, I'm afraid. But it is the mood music around the Brexit talks resuming in Brussels.

Meanwhile, at Westminster, there is "a new politics around Brexit" according to Phillip Blond, the director of the centre right Respublica think tank on BBC Breakfast this morning.

He was talking about Labour's policy shift: promising to stay in the single market and a customs union for a transition period after Brexit.

The reaction, on Radio 5 live's Your Call this morning, suggests the instinct of those on the winning side of a referendum is that such a vote should mark the end of the conversation on a subject: the people have spoken and we should leave, and leave properly.

And those on the losing side think it should mark the start of a conversation: about what leave should look like, or whether we should leave at all.

The UK is set to leave the EU by the end of March 2019, following last year's referendum vote.

In the first phase of negotiations, British and EU officials are meeting each month for four days in Brussels.

On Sunday, Labour said it would keep the UK in the single market and customs union for a transitional period after leaving the EU.

It said this was needed to avoid a "cliff edge" for the economy.

Meanwhile, the British Chambers of Commerce and the Association of German Chambers of Commerce have jointly pleaded with politicians to put economic interests first in their Brexit discussions.