During his Brussels meeting, Raab offered to meet Barnier "throughout the summer to intensify negotiations, to get some energy, get some drive and get some heat on them to make sure we can conclude this agreement in good time", the Brexit secretary said on Thursday afternoon.
"While the European Union is working day and night for a deal ensuring an orderly withdrawal, the UK's withdrawal will undoubtedly cause disruption, for example in business supply chains, whether or not there is a deal", the Commission said in a statement.
The EU executive also explicitly stated that "no progress" has been made in agreeing to a so-called "backstop" solution to the border issue on the island of Ireland, that would guarantee an invisible border at the EU's new external barrier between the Republic of Ireland and Northern Ireland.
But that did not stop Boris Johnson, the former foreign minister who quit last week over May's plan, from renewing his call for government to rethink it, saying "it is not too late to save Brexit".
"There is time. And, if the Prime Minister can fix that vision once again before us, then I believe she can deliver a great Brexit for Britain with a positive, self-confident approach that will unite this party, unite this House and unite this country as well".
However, the Chequers plan announced earlier this month sparked fury among Brexiteers as it would have locked the United Kingdom into a "single standards model" and a "common rulebook" for goods, effectively preventing Britain from adopting competing national standards.
Brexiteers believe that keeps Britain too close to the European Union, while pro-Europeans think it fails to protect the country's dominant services sector, among other gripes.
With the Conservative party deeply divided over Brexit, some at Westminster had expected Johnson to urge his colleagues to dislodge the prime minister.
Barnier has warned that it will be hard to conclude the negotiations by the October target, which would give parliaments on both sides time to endorse the deal.
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"We would like to have a withdrawal agreement as it [Brexit] would be a lot easier with it for both sides", one European Union official said on condition of anonymity.
The EU aims to give Britain a transition period until the end of 2020 to give businesses and others time to adjust to a post-Brexit future.
It also warned of possible disruption to transport, including aviation, as well as to the flow of goods as a result of customs checks.
One government minister said on condition of anonymity that the Conservatives had come to their senses and were backing her Brexit plan.
Varadkar said that even if there is a deal, Ireland will need 1,000 new customs officers and veterinary inspectors to deal with changes in trade rules with the UK.
David Davis, who resigned last week as Brexit Secretary in protest at her negotiating stance, challenged her to publish Whitehall work on a potential free trade deal based on those already secured by the EU.
He went onto say: "Everybody wants to have a withdrawal agreement, I'm still confident we can have one because the cost of not having a deal is disproportionally higher than having a withdrawal agreement".
And with the situation so "volatile" in London, the source said, they will "try not to put oil on the fire".