The EU insists on "sufficient progress" in sorting out the divorce - including a so-called Brexit bill for the U.K.'s outstanding financial obligations to the bloc - before moving on to discuss the terms of future economic relations.
Barnier's comments followed the third round of increasingly acrimonious Brexit negotiations in Brussels last week.
Britain would pay up to £17 billion a year to Brussels for three years after Brexit before ending payments ahead of the 2022 general election, the Times added.
Mr Davis said: "It's nonsense, the story is completely wrong".
"What he's [Barnier] concerned about of course is he's not getting the answer on money and they've set this up to try to create pressure on us on money", Davis said.
"Time is not running out".
"We'll still be paying something", he said, but not in the "medium to long-term" and that the department is "still going through legal arguments" and is doing so "line-by-line".
Theresa May's said it was time for Parliament to "play its part" by passing the European Union (Withdrawal) Bill at its second reading, but her deputy Damian Green issues a warning to colleagues planning to side with Labour, the SNP and Liberal Democrats to try keep the United Kingdom in the single market.
The latest round of discussions between Davis and the EU's chief negotiator Michel Barnier failed to make progress on the divorce bill prompting a tense press conference between the pair on Thursday (31 August) in which Barnier said there had been a lack of "decisive progress on any of the principal subjects".
He said Barnier "wants to put pressure on us, which is why the stance this week in the press conference - bluntly, I think it looked a bit silly because there plainly were things that we had achieved". The commission puts itself in a silly position when it says nothing has been done when really important things have.
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"If you take our fresh produce supply chains, for example, we put things on a lorry in Spain and it will arrive in a distribution centre somewhere in England, and it won't have gone through any border checks".
Comparing the financial settlement to a hotel bill, Davis said United Kingdom negotiators were checking the details, not making a counter offer.
"We're going through [the bill] line by line, and they're finding it hard because we've got good lawyers..."
Fox said businesses across Europe had told him they were keen to see more detail on what Britain's new relationship with the European Union would look like. Post-Brexit, the United Kingdom would "still be paying something, I suspect", Davis said, citing space agency research and the EU's Horizon 2020 scientific programme.
Coupe said he did not believe the government "fully recognised" the repercussions of supply chain disruption.
In remarks that are likely to enrage United Kingdom ministers, Barnier said Brexit would be "an educational process" for the British public who voted to leave the EU.
Quoting a 'source, ' the paper says that Davis and May are keeping the agreement until after the Conservative Party conference in October for fear of attacks from the party's Eurosceptic MPs.
Davis said all MPs had an interest in the bill succeeding. "Anybody, remainer or leaver, should support this bill", he said.
Mr Davis said: "We've said the era of big payments to the European Union is coming to an end - we'll still be paying something, I expect".