The EU fears controls on the 500-km (300 mile) Irish border with Northern Ireland would undermine the 1998 Good Friday Agreement which ended three decades of sectarian and political conflict that killed over 3,600 people.
Mr Barnier said the plans represented "a significant risk to the integrity of the single market" and warned: "We are one week away from the European Council summit and just a few weeks away from the date of October 31".
Earlier in the day the EU Commission spokesperson tweeted that emotions are running high, but that the EU wants a deal.
In a message to Mr Johnson, he said: "Prime Minister, we can't trust you not to break the law because you've got form". Varadkar told Irish reporters that he now thought it was possible for Britain to reach a deal to leave the European Union in an orderly way by an October 31 deadline.
He later added: "The reason the Prime Minister is meeting Leo Varadkar isn't simply to have a social conversation, they are seriously focused on trying to resolve this issue and trying to get a deal on which basis we can leave the European Union".
Ireland is by the biggest issue of disagreement.
Under the terms of the so-called Benn Act, the Prime Minister - or an official in his place - has to write to the European Union requesting an extension if no deal has been reached by 19 October. That deal was rejected three times by Britain's Parliament, largely because of lawmakers' opposition to provisions to ensure an open Irish border.
"So the next steps, of course, are for the United Kingdom government to engage with the European Union commission".
The two agreed to "reflect" further on their discussions, which concentrated on the challenges of customs and consent.
Back in Dublin the Taoiseach defended the Budget as a defence against a no-deal Brexit - and then insisted that, in preparing for the worst, he was not assuming that no deal was the only available outcome.
He proposes Northern Ireland stay aligned to the EU's single market but remain in a separate UK-wide customs territory, envisaging customs but no regulatory checks on the frontier.
There have been a number of headlines this week proclaiming that Brexit talks between the United Kingdom and EU are on the verge of collapse, fears of which have sent the Pound (GBP) plummeting to a one-month low against the Euro (EUR). The UK's central bank said that temporary permissions, legislation, and other preparations have been made to "ensure that households and businesses will be able to use existing and new services from European Union financial institutions".
"But we need to make sure that any such arrangements are workable", he said.
"The problem is that in Britain today we have a culture in which many leading politicians seem to think it is acceptable to flout constitutional conventions, while voters appear to be ready to endorse this behaviour at the ballot box", Blick added.