During the latest round of talks, David Frost insisted the EU negotiator would have to rethink his positions in order to end months of deadlock. The pair had dinner on Monday night in London before holding further “informal” discussions on the future relationship pact.
Officials on both sides were privately confident a deal can still be reached after good progress was made in talks last week.
But a UK Government spokesman said Brussels has not yet decided whether it is ready to accept the Prime Minister’s offer of a Canada-style free-trade agreement.
“We have had constructive discussions and both sides have shown pragmatism and willingness to move on difficult areas,” the spokesman said.
“What is needed now is internal reflection on the EU side on whether what they’re asking us is consistent with our status as an independent country.
“If they do that and come to the correct conclusion, we’ll be able to move forward.”
The two sides will now take a two-and-a-half week summer break before returning for the sixth round of negotiations in Brussels on August 17.
But Mr Frost and Mr Barnier are expected to continue working on their preparations throughout the downtime.
A European Commission spokeswoman said: “Michel Barnier is very busy, no holidays in sight.”
UK and EU officials will return to work on August 10 to ready themselves for the crunch sessions.
Mr Frost is confident he can find an agreement with Brussels by September, after negotiators missed the Prime Minister’s summer deadline.
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Diplomats from Lithuania and Hungary last week pressured their negotiator to drop his hardline demands in order to end the impasse.
France was infuriated by the non-coastal states’ attempt to stop the row over access ending chances of a Brexit deal.