The proposed plan will accept the need for both a regulatory border between the UK and Northern Ireland in the Irish Sea for four years, the Telegraph reports.
This will also include customs checks between Ulster and the Republic of Ireland.
It means Northern Ireland will leave the EU when the UK does on October 31, but will continue to follow single market rules for agriculture and industrial goods.
The proposal will essentially replace the Irish backstop, the Telegraph states.
However, the publication adds that it is unlikely the plan will go down well among EU leaders, as they would have to grant the UK ‘sweeping exemptions’ from the EU customs rules.
The radical new plan has been dubbed ‘two borders for four years’.
Boris will set out plans for his Brexit deal tomorrow at the Conservative Party conference, adding that there will be ‘no delay’ beyond the current October 31 deadline.
British officials have made clear to EU counterparts that the legal texts which will be presented to the EU this week are a final offer.
They have been told that unless Brussels is prepared to engage with the plan, there will be no more talks until after Brexit.
Boris will tell the conference: ‘Voters are desperate for us to focus on their other priorities – what people want, what leavers want, what remainers want, what the whole world wants – is to move on.
‘That is why we are coming out of the EU on October 31. Let’s get Brexit done – we can, we must and we will.’
A senior Number 10 official said: ‘The Government is either going to be negotiating a new deal or working on no deal – nobody will work on delay.
‘We will keep fighting to respect the biggest democratic vote in British history.’
Boris will avoid his regular Wednesday session of Prime Minster’s Questions to deliver the speech, after MPs refused to vote for a Commons recess while the Tory event takes place.
He is also expected to take shots at Jeremy Corbyn, accusing him of having a ‘Corbyn agenda’ that means staying in the EU past the deadline.
The Prime Minister will continue: ‘My friends, I am afraid that after three-and-a-half years people are beginning to feel that they are being taken for fools.
‘They are beginning to suspect that there are forces in this country that simply don’t want Brexit delivered at all.
‘And if they turn out to be right in that suspicion then I believe there will be grave consequences for trust in democracy.
‘Let’s get Brexit done on October 31 so in 2020 our country can move on.’Boris is unlikely to address recent allegations that he squeezed journalist Charlotte Edwardes’ thigh under the table at a dinner party in 1999.
The Prime Minister has already responded to the claims by stating that he has ‘no memory whatsoever’ of the incident and said it was ‘not true’.
He insisted: ‘It is not true, for all sorts of reasons, and I don’t wish as I say to minimise importance of subject.
‘I don’t wish to cast aspersions on the motives of anybody who makes this type of allegation but it is not true and what I want to do is focus on our domestic agenda.’
Boris has also been dogged by questions about his alleged affair with American entrepreneur Jennifer Arcuri and whether he granted her preferential treatment while Mayor of London.
He insisted nothing improper took place between the pair of them.