"I make no secret of the fact that to provide certainty we must talk about the future", Davis said, stressing his demand for trade talks.
Mr Juncker said: "The British are discovering, as we are, day after day new problems".
"The Brexit process will last longer than we anticipated", Juncker said.
The EU has told Britain that a summit next week will conclude that insufficient progress has been made in talks for Brussels to open negotiations on a future trade deal.
Stood alongside the UK's Brexit Secretary David Davis, Barnier said that Theresa May's Florence Speech had given "momentum to these negotiations" but that there was a "disturbing" deadlock over the size of the bill the United Kingdom pays to the EU.
"We can not find for the time being a real compromise as far as the remaining financial commitments of the United Kingdom are concerned".
He said: "My mandate looked ahead to these different stages.Slowly but surely, over the next few weeks, I will explore ways of getting out of this deadlock we find ourselves in on the financial issues with a view to making sufficient progress by the next European Council. I do not hate the British", he added.
Speaking about the financial settlement at a press conference in Brussels alongside UK Brexit Secertary David Davis on Thursday, Barnier said: "On this question we have reached a state of deadlock which is very disturbing for thousands of project promoters in Europe and it's disturbing also for taxpayers".
European Union leaders meeting in Brussels next week are expected to say that there is not likely to be enough progress until the next summit in December, but that the bloc should begin internal preparations for trade talks now. The Europeans have to be grateful for so many things Britain has brought to Europe before war, during war, after war. "But not they have to pay".
He said: "If you are sitting at a bar and ordering 28 beers and then suddenly some of your colleagues are leaving, it's ok but they have to pay, they have to pay".
His recommendation has made it all but certain that trade talks will be delayed at least until the end of the year, heightening pressure on the UK Government to make preparations for a possible "no-deal" Brexit. Some even say Britain should walk away from the negotiations if the European Union does not agree to start the next phase by the end of the year.