"If Jeremy Corbyn wants to be Prime Minister and to rebuild those public services he needs to win those voters by delivering a vote on the final deal and a chance for the country change course on Brexit".
After Mr Johnson's column claimed Mrs May's approach would leave the United Kingdom "lying flat on the canvass", the Number 10 spokesman said: "Boris Johnson resigned over Chequers, there's no new ideas in this article to respond to".
Downing Street's statement comes after Johnson used his Telegraph to accuse May of surrendering on Brexit.
Mr Barnier also offered further evidence he has ditched his previous deadline of October for tying up a Brexit deal, stating negotiations must now be concluded by mid-November.
And the spokesman added: "Boris Johnson resigned over Chequers".
He added: "There's no new ideas in this article to respond to".
"There is an urgent need to work on the text of a operational backstop and that's why I've asked Dominic and his team to provide us with the necessary data for technical work, which we need now, on the nature, location and methods of the checks that will be needed", Barnier said.More news: Tropical storm: Gordon expected to hit MS with hurricane force
Raab said he, along with Prime Minister Theresa May and other senior ministers have made visits across Europe, explaining Britain's Brexit proposals and making the case for what the government has put forward for a future relationship with the European Union after next March.
"The fix is in", said the former foreign secretary.
"Either of those outcomes could variously involve more serious economic consequences than anything seen so far, the fall of the Conservative government, a general election, and either the election of the dysfunctional Labour Party or another hung Parliament".
"The scandal is not that we have failed, but that we have not even tried", he said.
Writing in the Telegraph, Johnson accused May of getting "diddly squat" from negotiations and handing "victory" to the EU. He said that under the current plan, "we will remain in the European Union taxi; but this time locked in the boot, with absolutely no say on the destination".
Mr Barnier also declared the prime minister's plan to levy European Union tariffs on behalf of Brussels, in order to bypass the need for post-Brexit customs checks, would be an invitation to fraud.
The government says it is the only way to achieve Brexit without harming the economy.More news: Tropical Storm Gordon: What to expect in Alabama
Mr Johnson claimed the border issue was "fixable".
One of the country's biggest trade unions is calling for a public vote on the final Brexit deal, because it says promises made during the referendum are "not the reality we are facing". Parliament returns from its summer break on Tuesday.
EU's Barnier told a German newspaper on Sunday that he strongly opposed Britain's latest proposal.
Mr Davis said he would vote against the Prime Minister's Chequers plan in any Commons showdown as International Trade Secretary Liam Fox took a swipe at Chancellor Philip Hammond's no deal warnings.
Johnson fumed in a newspaper column that May's proposal to retain close economic ties with the European Union after Brexit would leave Britain locked in the trunk of a Brussels-driven vehicle with "no say on the destination".
As trade unionists, when we negotiate a pay deal with an employer, we go back to our members and ask them if they're happy with it.More news: Tropical Storm Gordon brings hurricane warning for parts of Gulf Coast