The EU agreed to a compromise on the backstop agreement to keep the border between Northern Ireland and Ireland last week, signalling they would accept keeping the whole of the United Kingdom in a customs union until both sides sign up to a deal that would prevent the need for a hard border.
The move could win over some opposition Labour MPs and increase the chances of getting such an agreement through the House of Commons.
Mrs May has reportedly secured concessions from the European Union to avoid a hard border in Ireland, the Sunday Times reports.
British Prime Minister Theresa May raised the possibility of a review mechanism for the backstop in a phone call on Monday with Varadkar that she had sought to update him on the current state of the talks, the Irish government said in a statement.
Theresa May will warn Brexiteer ministers on Tuesday that time is running out to clinch agreement with Brussels or face a no-deal departure from the European Union.
Under the banner headline "Revealed: Theresa May's secret Brexit deal", The Sunday Times said the European Union accepts that regulatory checks on goods can be carried out in factories and shops rather than at the border.
The opposition Labour Party has all but ruled out supporting any deal May reaches with Brussels, leaving the prime minister reliant on her slender parliamentary majority that comprises her own divided MPs and coalition partners the Democratic Unionist party.
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While European Council President Donald Tusk said in September that an extraordinary summit would be called for November 17-18 to finalize and formalize the EU-UK arrangement if sufficient progress in talks was reached, the EU leaders are now not planning this extraordinary summit.
Speaking to Sky News on Monday on whether Theresa May can get a Parliamentary majority to pass her Chequers Plan, which would leave the United Kingdom tied to numerous bloc's regulations, Mr Farage said: "In classic Brussels style, at the 11th hour there will be a deal, I've no doubt about that".
May's spokesman refused to confirm the report, calling it "speculation", according to Sky News.
Under the banner headline "May's Secret Brexit Deal", the newspaper said she's also on course to gain an agreement on a "future economic partnership" that will allow Britain to keep open the prospect of a similar free-trade accord to the one Canada has with the EU.
Farage added that if the British elite betrayed Brexit, he did not think there would be violence on the streets but that British politics would be shaken up and the Conservative Party would suffer badly.
But some of Britain's biggest corporate names have had enough of the government's handling of the negotiations.
"Both these options will further depress investment". But U.K. officials believe intensive negotiations will deliver the decisive step needed for an agreement within the next few weeks, according to two people familiar with the matter.