"I think it is possible to get a deal in November", he said.
In a major intervention on the controversial backstop, amid reports that the government plans to keep Northern Ireland in aspects of the European Union trade structures, Mr Davis said it was "pretty clear there is genuine and significant concern regarding the implications of any fresh backstop text".
Both the European Union and United Kingdom have agreed the need for a backstop, but not how it should operate.
The Sunday Times reports that the prime minister is close to a political deal on the United Kingdom and EU's so-called future economic partnership, which the newspaper says will "allow Britain to keep open the prospect of a free trade deal resembling that enjoyed by Canada".
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.
Reports at the weekend suggested that the EU is now ready to contemplate concessions which would keep all of the United Kingdom in a temporary customs union following the end of a transition period due to end on December 31 2020.
Mrs May is expected to brief the Cabinet on progress in talks when members gather for their weekly meeting at 10 Downing Street on Tuesday.
"The Democratic Unionist Party is rightly anxious about the future of the Union and many Conservatives are too".
Communities Secretary James Brokenshire, appearing on Sky's Ridge On Sunday, was asked if a deal was close.
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.
"There is no competitor worthy of replacing us", he added.
Irish foreign minister Simon Coveney responded by insisting that the European Union would never sign up to a backstop deal that gave the United Kingdom the ability to pull out any time it chose. The prime minister has been clear that we are making good progress on the future relationship and 95% of the withdrawal agreement is now settled.
Lidington also said London would stand by the written commitments it had already made on the backstop, which include the agreement that it would apply unless and until a better solution is found. "I certainly hope we are".
It came as more than 70 business leaders backed a referendum on the terms of the Brexit deal, warning that the United Kingdom faces "either a blindfold or a destructive hard Brexit" that would be bad for both firms and jobs.
Theresa May will be able to persuade her Cabinet to support her preferred Brexit deal - but only after more of her ministers resign, according to a government source.