The report claims an "all-UK customs deal" will be written into the legally binding withdrawal agreement, which would do away with the need for the controversial "backstop" arrangement agreed by the United Kingdom last December, which would see Northern Ireland remain in full alignment with the EU's single market and customs union rules in the event of a no-deal Brexit.
Here, Labour leader Brendan Howlin said he remained "very fearful" Irish interests could be sacrificed by Europe to secure a deal and said Ireland has to be "vigilant".
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".
Theresa May is said to have made a breakthrough on Brexit talks that could see her put forward a deal to her Cabinet as early as Tuesday.More news: Rooney to make England farewell appearance against USA
The backstop plan enraged the DUP, which is propping up May's Conservative government at Westminster, and the PM has since said she could "not accept" any deal that would require a customs border between the North and Great Britain.
The EU has proposed a backstop that would mean Northern Ireland staying in the EU customs union, large parts of the single market and the EU VAT system.
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.
Ahead of Mrs May's cabinet meeting, sources here said the situation was at its "most sensitive yet" and she will be "dancing on the head of a pin" to secure agreement on the deal in her government.More news: Police say 2 small planes crashed in Ottawa, no word on injuries
The publication states that the arrangement will will also include an "exit clause" that will aim at convincing Brexiteers that remaining in the customs union is only temporary.
A spokesperson said: "The prime minister has been clear that we are making good progress on the future relationship and 95pc of the Withdrawal Agreement is now settled and negotiations are ongoing".
Source close to European Union negotiator Michel Barnier revealed that a "major concession" had been made on the Irish border during a meeting in London last week that will see regulatory checks on goods carried out at factories and shops rather than being conducted at the border. "I certainly hope we are".
A special summit is understood to have been pencilled in for later this month to agree the final details of a deal.More news: Floyd Mayweather to fight Japanese kickboxing sensation on New Year's Eve
Both sides agree there must be no customs posts or other barriers that could disrupt businesses and residents or undermine Northern Ireland's peace.