The Brexit secretary is understood to have made his position clear in a meeting with Simon Coveney, the Irish foreign minister, last week.
An all-UK customs deal could be written into the withdrawal agreement, avoiding the need for a "backstop" for Northern Ireland.
Brexit is fraying Britain's relationship with Ireland, the country's prime minister Leo Varadkar has warned.
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".
Her plan would avoid the need for a hard border in Northern Ireland, which has always been the main sticking point in negotiations.
However, May's former Brexit Secretary has said she must publish legal advice on any deal ahead of a Commons vote, so MPs are fully briefed.
She is instead pursuing regulatory alignment with the EU and a "backstop" which would keep the entire country in the Customs Union as a "compromise" - but this would effectively preclude Britain from regaining an independent trade policy after Brexit.More news: Global smartphone shipments decline by 3% in Q3 of 2018
It comes after Raab reportedly told Coveney he wanted Britain to be able to quit any backstop arrangement after just three months, according to the Telegraph.
Writing in the Sunday Times, Mr Davis said: "It was an unwise decision to accept the EU's language on dealing with the Northern Ireland border".
The Sunday Times report said preparations for a final Brexit deal were "far more advanced than previously disclosed" and that May's agreement would satisfy both remain-voting Tories and the hardline Eurosceptics within her party.
At the same time, it reported that the prime minister was on course to agree a future economic partnership that would leave open the possibility of Canada-style free trade deal sought by Brexiteers.
"The prime minister has been clear that we are making good progress on the future relationship, and 95 percent of the withdrawal agreement is now settled and negotiations are ongoing".
Johnson believes the "rigmarole" around the border between the Republic of Ireland and the British province of Northern Ireland, which is relatively unimportant in economic terms, has been "designed by the European Union - and those in the United Kingdom who want us to remain in the European Union in all but name - in order to frustrate any future bid for independence". "I certainly hope we are".
It is speculated that May is hoping for enough progress in Brexit talks this week to secure a summit later this month in which the final details of a deal will be negotiated.